| L(s) = 1 | + (−0.732 + 2.73i)2-s + (7.34 + 12.7i)3-s + (−6.92 − 4i)4-s + (−10.3 − 10.3i)5-s + (−40.1 + 10.7i)6-s + (6.98 + 26.0i)7-s + (16 − 15.9i)8-s + (−67.4 + 116. i)9-s + (35.9 − 20.7i)10-s + (143. + 38.3i)11-s − 117. i·12-s + (−26.7 − 166. i)13-s − 76.2·14-s + (55.8 − 208. i)15-s + (31.9 + 55.4i)16-s + (122. + 70.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.683i)2-s + (0.816 + 1.41i)3-s + (−0.433 − 0.250i)4-s + (−0.415 − 0.415i)5-s + (−1.11 + 0.298i)6-s + (0.142 + 0.531i)7-s + (0.250 − 0.249i)8-s + (−0.832 + 1.44i)9-s + (0.359 − 0.207i)10-s + (1.18 + 0.316i)11-s − 0.816i·12-s + (−0.158 − 0.987i)13-s − 0.389·14-s + (0.248 − 0.925i)15-s + (0.124 + 0.216i)16-s + (0.423 + 0.244i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.752877 + 1.21229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.752877 + 1.21229i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.732 - 2.73i)T \) |
| 13 | \( 1 + (26.7 + 166. i)T \) |
| good | 3 | \( 1 + (-7.34 - 12.7i)T + (-40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (10.3 + 10.3i)T + 625iT^{2} \) |
| 7 | \( 1 + (-6.98 - 26.0i)T + (-2.07e3 + 1.20e3i)T^{2} \) |
| 11 | \( 1 + (-143. - 38.3i)T + (1.26e4 + 7.32e3i)T^{2} \) |
| 17 | \( 1 + (-122. - 70.6i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-662. + 177. i)T + (1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (732. - 423. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-113. - 195. i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (997. + 997. i)T + 9.23e5iT^{2} \) |
| 37 | \( 1 + (26.7 + 7.16i)T + (1.62e6 + 9.37e5i)T^{2} \) |
| 41 | \( 1 + (-32.0 + 119. i)T + (-2.44e6 - 1.41e6i)T^{2} \) |
| 43 | \( 1 + (2.83e3 + 1.63e3i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-405. + 405. i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + 2.42e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-751. - 2.80e3i)T + (-1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (1.21e3 - 2.11e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (215. - 804. i)T + (-1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 + (-6.05e3 + 1.62e3i)T + (2.20e7 - 1.27e7i)T^{2} \) |
| 73 | \( 1 + (-1.36e3 + 1.36e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 - 5.79e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (2.99e3 + 2.99e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + (1.04e3 + 280. i)T + (5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (7.49e3 - 2.00e3i)T + (7.66e7 - 4.42e7i)T^{2} \) |
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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
−16.65303357716570227376311605558, −15.70649403692000046688977260021, −14.98062719188576087132719075030, −13.91478224663940552352087255914, −11.90271303000443908553454403750, −9.986559330694054101298765364148, −9.087071939058310377625213462370, −7.87224494698402402447965729596, −5.33038203541632794091184129268, −3.75596788602024614367664358052,
1.44596788556305435584160845574, 3.48577853586382841255474497752, 6.86103771086488754992419303461, 7.986797369859036338912235741047, 9.463548471532753765962419340019, 11.48509955232072546710380001634, 12.32842671667810978301093255399, 13.98404846246052171519000886838, 14.23992620208760133363715631509, 16.57009956932081517491593065526
