| L(s) = 1 | − 3·3-s + 8·5-s + 7-s + 8·9-s + 11-s + 3·13-s − 24·15-s − 17-s − 9·19-s − 3·21-s − 9·23-s + 20·25-s − 20·27-s − 5·29-s + 4·31-s − 3·33-s + 8·35-s − 9·39-s + 19·41-s − 13·43-s + 64·45-s + 5·47-s + 2·49-s + 3·51-s − 5·53-s + 8·55-s + 27·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 3.57·5-s + 0.377·7-s + 8/3·9-s + 0.301·11-s + 0.832·13-s − 6.19·15-s − 0.242·17-s − 2.06·19-s − 0.654·21-s − 1.87·23-s + 4·25-s − 3.84·27-s − 0.928·29-s + 0.718·31-s − 0.522·33-s + 1.35·35-s − 1.44·39-s + 2.96·41-s − 1.98·43-s + 9.54·45-s + 0.729·47-s + 2/7·49-s + 0.420·51-s − 0.686·53-s + 1.07·55-s + 3.57·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.288129457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288129457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 + p T + T^{2} - T^{3} + 4 T^{4} - p T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 7 | $C_2^2:C_4$ | \( 1 - T - T^{2} - 17 T^{3} + 64 T^{4} - 17 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 - T - 5 T^{2} - 29 T^{3} + 144 T^{4} - 29 p T^{5} - 5 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 3 T - 9 T^{2} + 11 T^{3} + 144 T^{4} + 11 p T^{5} - 9 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + T - T^{2} - 53 T^{3} + 104 T^{4} - 53 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 9 T + 17 T^{2} - 3 T^{3} + 100 T^{4} - 3 p T^{5} + 17 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 9 T + p T^{2} - 135 T^{3} - 1424 T^{4} - 135 p T^{5} + p^{3} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 5 T + 11 T^{2} + 195 T^{3} + 1736 T^{4} + 195 p T^{5} + 11 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 19 T + 95 T^{2} + 599 T^{3} - 8776 T^{4} + 599 p T^{5} + 95 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2:C_4$ | \( 1 + 13 T + 21 T^{2} - 511 T^{3} - 4756 T^{4} - 511 p T^{5} + 21 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 5 T - 7 T^{2} - 285 T^{3} + 3644 T^{4} - 285 p T^{5} - 7 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 5 T + 7 T^{2} - 245 T^{3} - 1056 T^{4} - 245 p T^{5} + 7 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 13 T + 5 T^{2} + 687 T^{3} - 5716 T^{4} + 687 p T^{5} + 5 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 15 T + 19 T^{2} + 15 p T^{3} - 12104 T^{4} + 15 p^{2} T^{5} + 19 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 9 T + 63 T^{2} + 155 T^{3} - 2184 T^{4} + 155 p T^{5} + 63 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 7 T - 45 T^{2} - 167 T^{3} + 7784 T^{4} - 167 p T^{5} - 45 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + T - 7 T^{2} + 645 T^{3} + 6356 T^{4} + 645 p T^{5} - 7 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 17 T + 95 T^{2} + 1187 T^{3} + 18584 T^{4} + 1187 p T^{5} + 95 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 13 T + 147 T^{2} + 1295 T^{3} + 4256 T^{4} + 1295 p T^{5} + 147 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05290589470534174010775603961, −9.546348042336218748692361613761, −9.535797367254221724560361676584, −9.362415443689986268240072028782, −9.076835067710406441662938057508, −8.431711620350076419339393590975, −8.172950037806157566288035276702, −7.87911980526500187428491479122, −7.70331379204370641451767953858, −6.77091952332651142697654887407, −6.68493089377561079420815357821, −6.61985374011884888005382683946, −6.36094032349605316617161456445, −5.73642177353656937848152978897, −5.68135150162046020823927506609, −5.56067872467934032443089727442, −5.54138470280814528757436015335, −4.71093931147597234565160447130, −4.28289565152854067390969419466, −3.93669002607716546537229718179, −3.87576542617980737553267830864, −2.37244873422439160351656115053, −2.23569078335373275558135236022, −1.85733808137285213790918834189, −1.42340413758015106948241921972,
1.42340413758015106948241921972, 1.85733808137285213790918834189, 2.23569078335373275558135236022, 2.37244873422439160351656115053, 3.87576542617980737553267830864, 3.93669002607716546537229718179, 4.28289565152854067390969419466, 4.71093931147597234565160447130, 5.54138470280814528757436015335, 5.56067872467934032443089727442, 5.68135150162046020823927506609, 5.73642177353656937848152978897, 6.36094032349605316617161456445, 6.61985374011884888005382683946, 6.68493089377561079420815357821, 6.77091952332651142697654887407, 7.70331379204370641451767953858, 7.87911980526500187428491479122, 8.172950037806157566288035276702, 8.431711620350076419339393590975, 9.076835067710406441662938057508, 9.362415443689986268240072028782, 9.535797367254221724560361676584, 9.546348042336218748692361613761, 10.05290589470534174010775603961
