| L(s) = 1 | + 3·5-s − 3·7-s − 3·9-s + 3·11-s + 5·13-s + 9·23-s + 25-s + 3·29-s − 9·31-s − 9·35-s + 4·37-s − 9·41-s + 9·43-s − 9·45-s + 3·47-s − 49-s + 9·55-s − 3·59-s + 61-s + 9·63-s + 15·65-s + 15·67-s − 24·71-s − 4·73-s − 9·77-s − 15·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s − 9-s + 0.904·11-s + 1.38·13-s + 1.87·23-s + 1/5·25-s + 0.557·29-s − 1.61·31-s − 1.52·35-s + 0.657·37-s − 1.40·41-s + 1.37·43-s − 1.34·45-s + 0.437·47-s − 1/7·49-s + 1.21·55-s − 0.390·59-s + 0.128·61-s + 1.13·63-s + 1.86·65-s + 1.83·67-s − 2.84·71-s − 0.468·73-s − 1.02·77-s − 1.68·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.322090324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322090324\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.24115208490608309117435303920, −13.22180926602360114803102296989, −12.40152109552675379619288166045, −11.87905048699547129785020985695, −11.20291703544438106782868112281, −10.84736513457373417686869429970, −10.34028644547091426521377671830, −9.511677227347317335941811743246, −9.312944172572840639402570621401, −8.885870633594408685217822171037, −8.361474323775439297448556447842, −7.39513206385297268528046215994, −6.53807913591060777441140676272, −6.46462364300244324935226542980, −5.65049118824998910766678737467, −5.39849902506889789807652657025, −4.14480418184802737496489738642, −3.35383496393844425103258497966, −2.71520314460127089964457787895, −1.43834980079011542247287906633,
1.43834980079011542247287906633, 2.71520314460127089964457787895, 3.35383496393844425103258497966, 4.14480418184802737496489738642, 5.39849902506889789807652657025, 5.65049118824998910766678737467, 6.46462364300244324935226542980, 6.53807913591060777441140676272, 7.39513206385297268528046215994, 8.361474323775439297448556447842, 8.885870633594408685217822171037, 9.312944172572840639402570621401, 9.511677227347317335941811743246, 10.34028644547091426521377671830, 10.84736513457373417686869429970, 11.20291703544438106782868112281, 11.87905048699547129785020985695, 12.40152109552675379619288166045, 13.22180926602360114803102296989, 13.24115208490608309117435303920
