L(s) = 1 | + (−3.54 + 2.04i)3-s + (15.5 + 27.0i)5-s + 2.67·7-s + (−32.1 + 55.6i)9-s − 42.3·11-s + (113. + 65.6i)13-s + (−110. − 63.8i)15-s + (143. + 247. i)17-s + (−326. + 153. i)19-s + (−9.50 + 5.48i)21-s + (158. − 273. i)23-s + (−173. + 300. i)25-s − 594. i·27-s + (−188. − 108. i)29-s + 475. i·31-s + ⋯ |
L(s) = 1 | + (−0.394 + 0.227i)3-s + (0.623 + 1.08i)5-s + 0.0546·7-s + (−0.396 + 0.686i)9-s − 0.349·11-s + (0.673 + 0.388i)13-s + (−0.491 − 0.283i)15-s + (0.495 + 0.858i)17-s + (−0.904 + 0.426i)19-s + (−0.0215 + 0.0124i)21-s + (0.299 − 0.517i)23-s + (−0.277 + 0.480i)25-s − 0.815i·27-s + (−0.224 − 0.129i)29-s + 0.495i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.180i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.116638665\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.116638665\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (326. - 153. i)T \) |
good | 3 | \( 1 + (3.54 - 2.04i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-15.5 - 27.0i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 - 2.67T + 2.40e3T^{2} \) |
| 11 | \( 1 + 42.3T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-113. - 65.6i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (-143. - 247. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 23 | \( 1 + (-158. + 273. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (188. + 108. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 - 475. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.74e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (104. - 60.5i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.23e3 - 2.14e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (168. - 292. i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (3.65e3 + 2.11e3i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (4.91e3 - 2.83e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.94e3 - 3.37e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (2.80e3 + 1.61e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (4.05e3 - 2.33e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-803. - 1.39e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-2.90e3 + 1.68e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 6.54e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-3.17e3 - 1.83e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-4.87e3 + 2.81e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−11.05126545364598111929677707895, −10.77764826365338097968837690517, −9.942762360755624035500832005756, −8.661426248706559080730731060120, −7.66103625189756977199498562070, −6.36627858976013122505063785888, −5.81982343425238140994521647201, −4.44799275395908739925763318957, −3.02363449413613349988805208936, −1.82981190950712868585995716263,
0.35548331955496102214424331738, 1.48063275133485385875827019359, 3.16570066797020616339116297322, 4.75146814378429760538272573946, 5.61792564380675130724880022029, 6.48142664339527121593735452502, 7.83346363831733650062066273070, 8.896948692621825719613462672718, 9.495952648036000180906780959783, 10.72423265815214161133344010745