| L(s) = 1 | + (−4.18 + 1.12i)2-s + (0.900 − 3.36i)3-s + (−11.4 + 6.59i)4-s + (36.1 − 42.6i)5-s + 15.0i·6-s + (−63.7 + 112. i)7-s + (138. − 138. i)8-s + (199. + 115. i)9-s + (−103. + 219. i)10-s + (193. + 334. i)11-s + (11.8 + 44.3i)12-s + (481. + 481. i)13-s + (140. − 544. i)14-s + (−110. − 159. i)15-s + (−214. + 370. i)16-s + (1.19e3 + 320. i)17-s + ⋯ |
| L(s) = 1 | + (−0.740 + 0.198i)2-s + (0.0577 − 0.215i)3-s + (−0.356 + 0.206i)4-s + (0.646 − 0.762i)5-s + 0.171i·6-s + (−0.491 + 0.870i)7-s + (0.765 − 0.765i)8-s + (0.822 + 0.475i)9-s + (−0.327 + 0.693i)10-s + (0.481 + 0.833i)11-s + (0.0238 + 0.0888i)12-s + (0.789 + 0.789i)13-s + (0.191 − 0.742i)14-s + (−0.127 − 0.183i)15-s + (−0.209 + 0.362i)16-s + (1.00 + 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.703i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.711 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.01290 + 0.416230i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01290 + 0.416230i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-36.1 + 42.6i)T \) |
| 7 | \( 1 + (63.7 - 112. i)T \) |
| good | 2 | \( 1 + (4.18 - 1.12i)T + (27.7 - 16i)T^{2} \) |
| 3 | \( 1 + (-0.900 + 3.36i)T + (-210. - 121.5i)T^{2} \) |
| 11 | \( 1 + (-193. - 334. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-481. - 481. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-1.19e3 - 320. i)T + (1.22e6 + 7.09e5i)T^{2} \) |
| 19 | \( 1 + (495. - 858. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.21e3 + 4.55e3i)T + (-5.57e6 + 3.21e6i)T^{2} \) |
| 29 | \( 1 - 5.44e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (1.45e3 - 841. i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-1.39e4 + 3.73e3i)T + (6.00e7 - 3.46e7i)T^{2} \) |
| 41 | \( 1 - 1.90e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (2.71e3 - 2.71e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (1.46e3 + 5.46e3i)T + (-1.98e8 + 1.14e8i)T^{2} \) |
| 53 | \( 1 + (6.66e3 + 1.78e3i)T + (3.62e8 + 2.09e8i)T^{2} \) |
| 59 | \( 1 + (-3.18e3 - 5.52e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (3.59e4 + 2.07e4i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (3.02e3 - 1.12e4i)T + (-1.16e9 - 6.75e8i)T^{2} \) |
| 71 | \( 1 - 3.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.07e3 + 7.73e3i)T + (-1.79e9 - 1.03e9i)T^{2} \) |
| 79 | \( 1 + (1.55e3 + 899. i)T + (1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (4.11e4 + 4.11e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + (-3.66e4 + 6.34e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (5.87e4 - 5.87e4i)T - 8.58e9iT^{2} \) |
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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
−16.30455734122069722915612692320, −14.43146408353382643013311400098, −12.97257103210981260869948244312, −12.42544006360227535826476666012, −10.12518987964430867661786650650, −9.236457241079015236721471240918, −8.160190790885513079939089412426, −6.43928719044975910755525689933, −4.47585940596976960544393579179, −1.54303542876266442276899486640,
1.00209548562979374268295960701, 3.65790559208377669893005934892, 5.95355827851617299524444800727, 7.60701038330474009704512080864, 9.394337163985442016003549011222, 10.11087597061225399144774740785, 11.13739846776155476099988235582, 13.30745771046733802451910489685, 13.98955613846096394112228731440, 15.42738092468878846825439123720
