| L(s) = 1 | + (1.02 − 0.162i)2-s + (−4.17 + 8.19i)3-s + (−14.1 + 4.60i)4-s + (1.67 + 24.9i)5-s + (−2.96 + 9.11i)6-s + (26.8 − 26.8i)7-s + (−28.6 + 14.6i)8-s + (−2.13 − 2.94i)9-s + (5.78 + 25.3i)10-s + (62.1 + 45.1i)11-s + (21.4 − 135. i)12-s + (−45.4 − 7.19i)13-s + (23.2 − 32.0i)14-s + (−211. − 90.4i)15-s + (165. − 120. i)16-s + (−49.6 − 97.3i)17-s + ⋯ |
| L(s) = 1 | + (0.257 − 0.0407i)2-s + (−0.464 + 0.910i)3-s + (−0.886 + 0.288i)4-s + (0.0668 + 0.997i)5-s + (−0.0822 + 0.253i)6-s + (0.548 − 0.548i)7-s + (−0.448 + 0.228i)8-s + (−0.0263 − 0.0363i)9-s + (0.0578 + 0.253i)10-s + (0.513 + 0.373i)11-s + (0.149 − 0.941i)12-s + (−0.268 − 0.0425i)13-s + (0.118 − 0.163i)14-s + (−0.939 − 0.402i)15-s + (0.648 − 0.470i)16-s + (−0.171 − 0.336i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.627704 + 0.871403i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.627704 + 0.871403i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.67 - 24.9i)T \) |
| good | 2 | \( 1 + (-1.02 + 0.162i)T + (15.2 - 4.94i)T^{2} \) |
| 3 | \( 1 + (4.17 - 8.19i)T + (-47.6 - 65.5i)T^{2} \) |
| 7 | \( 1 + (-26.8 + 26.8i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 + (-62.1 - 45.1i)T + (4.52e3 + 1.39e4i)T^{2} \) |
| 13 | \( 1 + (45.4 + 7.19i)T + (2.71e4 + 8.82e3i)T^{2} \) |
| 17 | \( 1 + (49.6 + 97.3i)T + (-4.90e4 + 6.75e4i)T^{2} \) |
| 19 | \( 1 + (-372. - 120. i)T + (1.05e5 + 7.66e4i)T^{2} \) |
| 23 | \( 1 + (-99.0 - 625. i)T + (-2.66e5 + 8.64e4i)T^{2} \) |
| 29 | \( 1 + (-1.10e3 + 358. i)T + (5.72e5 - 4.15e5i)T^{2} \) |
| 31 | \( 1 + (458. - 1.40e3i)T + (-7.47e5 - 5.42e5i)T^{2} \) |
| 37 | \( 1 + (-304. + 1.92e3i)T + (-1.78e6 - 5.79e5i)T^{2} \) |
| 41 | \( 1 + (-1.93e3 + 1.40e3i)T + (8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 + (1.74e3 + 1.74e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (300. + 152. i)T + (2.86e6 + 3.94e6i)T^{2} \) |
| 53 | \( 1 + (-1.52e3 + 2.98e3i)T + (-4.63e6 - 6.38e6i)T^{2} \) |
| 59 | \( 1 + (-1.61e3 - 2.22e3i)T + (-3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 + (4.07e3 + 2.96e3i)T + (4.27e6 + 1.31e7i)T^{2} \) |
| 67 | \( 1 + (-2.24e3 - 4.40e3i)T + (-1.18e7 + 1.63e7i)T^{2} \) |
| 71 | \( 1 + (1.01e3 + 3.12e3i)T + (-2.05e7 + 1.49e7i)T^{2} \) |
| 73 | \( 1 + (-417. - 2.63e3i)T + (-2.70e7 + 8.77e6i)T^{2} \) |
| 79 | \( 1 + (-4.16e3 + 1.35e3i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (-1.10e4 + 5.61e3i)T + (2.78e7 - 3.83e7i)T^{2} \) |
| 89 | \( 1 + (6.13e3 - 8.44e3i)T + (-1.93e7 - 5.96e7i)T^{2} \) |
| 97 | \( 1 + (427. + 217. i)T + (5.20e7 + 7.16e7i)T^{2} \) |
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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
−17.36794765085269855095030246722, −15.91339993318980251615842621499, −14.55421303677528836499898594787, −13.73163498784313724707807681959, −11.87556504507252124731169814077, −10.58025179652754237231261106584, −9.480352014066283693689441804365, −7.44559743663864322955951284202, −5.20103596967193184645234969594, −3.78851761583149257257852456529,
0.982689660983934285779508312659, 4.71640376212443290624949480874, 6.12483308454082827719279613064, 8.253144273760978719665249071629, 9.481131401943867464583085736425, 11.73005348189909360814153411062, 12.66104807312017153630011087440, 13.65621192769545301294341447417, 15.00856088469913216941487394434, 16.72531740026286689687768729408
