| L(s) = 1 | + (−4.27 − 7.39i)5-s + (12.5 + 13.5i)7-s + (−27.4 − 15.8i)11-s − 9.92i·13-s + (−63.7 + 110. i)17-s + (100. − 58.2i)19-s + (55.8 − 32.2i)23-s + (26.0 − 45.0i)25-s + 113. i·29-s + (−6.33 − 3.65i)31-s + (46.8 − 151. i)35-s + (−184. − 319. i)37-s − 211.·41-s + 432.·43-s + (−200. − 346. i)47-s + ⋯ |
| L(s) = 1 | + (−0.382 − 0.661i)5-s + (0.679 + 0.733i)7-s + (−0.752 − 0.434i)11-s − 0.211i·13-s + (−0.909 + 1.57i)17-s + (1.21 − 0.703i)19-s + (0.506 − 0.292i)23-s + (0.208 − 0.360i)25-s + 0.723i·29-s + (−0.0367 − 0.0211i)31-s + (0.226 − 0.729i)35-s + (−0.820 − 1.42i)37-s − 0.807·41-s + 1.53·43-s + (−0.620 − 1.07i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0818 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0818 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.386039259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.386039259\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-12.5 - 13.5i)T \) |
| good | 5 | \( 1 + (4.27 + 7.39i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (27.4 + 15.8i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 9.92iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (63.7 - 110. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-100. + 58.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-55.8 + 32.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 113. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (6.33 + 3.65i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (184. + 319. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 211.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 432.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (200. + 346. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-121. - 70.3i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-259. + 449. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-23.5 + 13.6i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (68.3 - 118. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 604. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (41.9 + 24.2i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (415. + 719. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 37.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + (235. + 407. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 522. iT - 9.12e5T^{2} \) |
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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
−8.998572278820343886250252460457, −8.634283528173216553740243110381, −7.899932942117326617201775135038, −6.89772510117680725031825917192, −5.66681346359956993772463695281, −5.11006014488457110029607761064, −4.13032920989154581468563590803, −2.89199314894344577176325892676, −1.73371332019896879629343539303, −0.38480929888588060042711246001,
1.08748203715038715409827658256, 2.49633970630991971043207702490, 3.47168544890130885050137026217, 4.62768710753190703415522682999, 5.27852476553272016386054482382, 6.68328489412731174658450084415, 7.39510019939233955997983988316, 7.82663975372402623851413568610, 9.025614472322328652146418315148, 9.911018979817800244384202024950
