| L(s) = 1 | + (1.5 + 2.59i)3-s + (0.723 − 1.25i)5-s + (12.3 + 13.7i)7-s + (−4.5 + 7.79i)9-s + (23.0 + 39.9i)11-s − 32.2·13-s + 4.33·15-s + (−38.8 − 67.3i)17-s + (6.33 − 10.9i)19-s + (−17.1 + 52.8i)21-s + (−50.4 + 87.4i)23-s + (61.4 + 106. i)25-s − 27·27-s + 213.·29-s + (21.0 + 36.4i)31-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (0.0646 − 0.112i)5-s + (0.669 + 0.743i)7-s + (−0.166 + 0.288i)9-s + (0.632 + 1.09i)11-s − 0.687·13-s + 0.0746·15-s + (−0.554 − 0.961i)17-s + (0.0764 − 0.132i)19-s + (−0.178 + 0.549i)21-s + (−0.457 + 0.792i)23-s + (0.491 + 0.851i)25-s − 0.192·27-s + 1.36·29-s + (0.121 + 0.211i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.919003843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.919003843\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 7 | \( 1 + (-12.3 - 13.7i)T \) |
| good | 5 | \( 1 + (-0.723 + 1.25i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-23.0 - 39.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 32.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (38.8 + 67.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-6.33 + 10.9i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (50.4 - 87.4i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 213.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-21.0 - 36.4i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (155. - 268. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 44.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 381.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (179. - 310. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (92.4 + 160. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-227. - 393. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (5.92 - 10.2i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-295. - 511. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 494.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (487. + 844. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-149. + 259. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-347. + 602. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 481.T + 9.12e5T^{2} \) |
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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
−11.64170083575825388445436769669, −10.29834825869108086819897584418, −9.474243179040710709625713560300, −8.779450737672474717280306853353, −7.67199594756388531479060761453, −6.64351724211696323477230640345, −5.11689227084948775615102871650, −4.59096663628232547520309070751, −2.97340096199857233941408067204, −1.71557323707617545908192636483,
0.66437867742761939625950067195, 2.08409268195808082830657255970, 3.57534598590018493128763666876, 4.71733311671190851937317336996, 6.19063266516410592112533293720, 6.95413591320751337221719221357, 8.183571422819845656667392549229, 8.642959017840140815885490240023, 10.07178423350887390611757542040, 10.84394624389041067368139796540
