| L(s) = 1 | + 2-s + 4-s + 2·5-s − 5·7-s + 3·8-s + 2·10-s + 11-s − 2·13-s − 5·14-s + 16-s − 17-s − 2·19-s + 2·20-s + 22-s + 9·23-s + 3·25-s − 2·26-s − 5·28-s − 6·29-s + 12·31-s − 32-s − 34-s − 10·35-s − 9·37-s − 2·38-s + 6·40-s + 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 1.88·7-s + 1.06·8-s + 0.632·10-s + 0.301·11-s − 0.554·13-s − 1.33·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s + 0.447·20-s + 0.213·22-s + 1.87·23-s + 3/5·25-s − 0.392·26-s − 0.944·28-s − 1.11·29-s + 2.15·31-s − 0.176·32-s − 0.171·34-s − 1.69·35-s − 1.47·37-s − 0.324·38-s + 0.948·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.854592543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.854592543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 25 T + 294 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 122 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 160 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T - 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.88646095615845168123369960683, −10.28573085179408317441480696404, −10.23759678723276250291360854061, −9.653358587885398934409783415698, −9.207299800836712096569848841156, −8.960271906157408174203070085830, −8.350910787573878956863667662519, −7.63962013293101031886046918865, −6.95240748263318397239298761811, −6.74099003219979492542022846742, −6.65114742430859964268471475635, −5.88757229265970843563261417573, −5.26215589933041183223434257422, −5.12356601790988876342595548419, −4.16458282532561182364840712696, −3.89653316860569168523808732707, −3.05987472392565070404397409404, −2.61435092927216633180023278223, −2.03545650757098222040794671648, −0.865108591650197847522266697336,
0.865108591650197847522266697336, 2.03545650757098222040794671648, 2.61435092927216633180023278223, 3.05987472392565070404397409404, 3.89653316860569168523808732707, 4.16458282532561182364840712696, 5.12356601790988876342595548419, 5.26215589933041183223434257422, 5.88757229265970843563261417573, 6.65114742430859964268471475635, 6.74099003219979492542022846742, 6.95240748263318397239298761811, 7.63962013293101031886046918865, 8.350910787573878956863667662519, 8.960271906157408174203070085830, 9.207299800836712096569848841156, 9.653358587885398934409783415698, 10.23759678723276250291360854061, 10.28573085179408317441480696404, 10.88646095615845168123369960683
