L(s) = 1 | − 4·2-s + 14·3-s + 16·4-s − 56·6-s + 158·7-s − 64·8-s − 47·9-s − 148·11-s + 224·12-s + 684·13-s − 632·14-s + 256·16-s + 2.04e3·17-s + 188·18-s + 2.22e3·19-s + 2.21e3·21-s + 592·22-s − 1.24e3·23-s − 896·24-s − 2.73e3·26-s − 4.06e3·27-s + 2.52e3·28-s − 270·29-s − 2.04e3·31-s − 1.02e3·32-s − 2.07e3·33-s − 8.19e3·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.898·3-s + 1/2·4-s − 0.635·6-s + 1.21·7-s − 0.353·8-s − 0.193·9-s − 0.368·11-s + 0.449·12-s + 1.12·13-s − 0.861·14-s + 1/4·16-s + 1.71·17-s + 0.136·18-s + 1.41·19-s + 1.09·21-s + 0.260·22-s − 0.491·23-s − 0.317·24-s − 0.793·26-s − 1.07·27-s + 0.609·28-s − 0.0596·29-s − 0.382·31-s − 0.176·32-s − 0.331·33-s − 1.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.805985935\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.805985935\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 14 T + p^{5} T^{2} \) |
| 7 | \( 1 - 158 T + p^{5} T^{2} \) |
| 11 | \( 1 + 148 T + p^{5} T^{2} \) |
| 13 | \( 1 - 684 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2048 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2220 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1246 T + p^{5} T^{2} \) |
| 29 | \( 1 + 270 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2048 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4372 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2398 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2294 T + p^{5} T^{2} \) |
| 47 | \( 1 + 10682 T + p^{5} T^{2} \) |
| 53 | \( 1 - 2964 T + p^{5} T^{2} \) |
| 59 | \( 1 + 39740 T + p^{5} T^{2} \) |
| 61 | \( 1 + 42298 T + p^{5} T^{2} \) |
| 67 | \( 1 - 32098 T + p^{5} T^{2} \) |
| 71 | \( 1 + 4248 T + p^{5} T^{2} \) |
| 73 | \( 1 - 30104 T + p^{5} T^{2} \) |
| 79 | \( 1 - 35280 T + p^{5} T^{2} \) |
| 83 | \( 1 + 27826 T + p^{5} T^{2} \) |
| 89 | \( 1 + 85210 T + p^{5} T^{2} \) |
| 97 | \( 1 + 97232 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.48894733048733978744819970322, −13.77684563564674653682899144955, −11.98569556409143960566713686926, −10.91476061918097024736032333103, −9.536697094708796313829082460504, −8.291327428170497872094901139815, −7.68859691366184980347896374583, −5.54626952739052738478306820479, −3.28499825800365520275696337731, −1.46276566744046286466573347456,
1.46276566744046286466573347456, 3.28499825800365520275696337731, 5.54626952739052738478306820479, 7.68859691366184980347896374583, 8.291327428170497872094901139815, 9.536697094708796313829082460504, 10.91476061918097024736032333103, 11.98569556409143960566713686926, 13.77684563564674653682899144955, 14.48894733048733978744819970322