| L(s) = 1 | − 15.1·3-s + 38.6·5-s − 225.·7-s − 14.3·9-s − 598.·11-s − 286.·13-s − 583.·15-s + 236.·17-s + 1.04e3·19-s + 3.40e3·21-s + 4.65e3·23-s − 1.63e3·25-s + 3.89e3·27-s − 632.·29-s + 7.78e3·31-s + 9.04e3·33-s − 8.70e3·35-s − 7.48e3·37-s + 4.33e3·39-s + 1.84e4·41-s + 7.30e3·43-s − 554.·45-s + 2.23e4·47-s + 3.39e4·49-s − 3.58e3·51-s − 2.67e4·53-s − 2.30e4·55-s + ⋯ |
| L(s) = 1 | − 0.969·3-s + 0.690·5-s − 1.73·7-s − 0.0591·9-s − 1.49·11-s − 0.470·13-s − 0.670·15-s + 0.198·17-s + 0.666·19-s + 1.68·21-s + 1.83·23-s − 0.522·25-s + 1.02·27-s − 0.139·29-s + 1.45·31-s + 1.44·33-s − 1.20·35-s − 0.899·37-s + 0.456·39-s + 1.71·41-s + 0.602·43-s − 0.0408·45-s + 1.47·47-s + 2.02·49-s − 0.192·51-s − 1.30·53-s − 1.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 15.1T + 243T^{2} \) |
| 5 | \( 1 - 38.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 225.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 598.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 286.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 236.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.65e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 632.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.48e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.23e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.67e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.09e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.98e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.37e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.17e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.58e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.03e5T + 8.58e9T^{2} \) |
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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
−9.150359521831876746350803774447, −7.76904903973517672183316206580, −6.91736832669330339008266097381, −6.05597610750917301318662674014, −5.56998905473457067849704200579, −4.71457716825795028559227890607, −3.09534913969894837087127539661, −2.62697652169961750921062657384, −0.850688809430662097741284181111, 0,
0.850688809430662097741284181111, 2.62697652169961750921062657384, 3.09534913969894837087127539661, 4.71457716825795028559227890607, 5.56998905473457067849704200579, 6.05597610750917301318662674014, 6.91736832669330339008266097381, 7.76904903973517672183316206580, 9.150359521831876746350803774447
