| L(s) = 1 | + (−0.647 + 0.762i)2-s + (−0.796 − 0.605i)3-s + (−0.161 − 0.986i)4-s + (0.140 + 0.265i)5-s + (0.976 − 0.214i)6-s + (−0.694 − 0.0754i)7-s + (0.856 + 0.515i)8-s + (0.267 + 0.963i)9-s + (−0.293 − 0.0646i)10-s + (1.30 + 0.441i)11-s + (−0.468 + 0.883i)12-s + (1.02 − 3.67i)13-s + (0.506 − 0.480i)14-s + (0.0486 − 0.296i)15-s + (−0.947 + 0.319i)16-s + (4.84 − 0.527i)17-s + ⋯ |
| L(s) = 1 | + (−0.457 + 0.538i)2-s + (−0.459 − 0.349i)3-s + (−0.0808 − 0.493i)4-s + (0.0629 + 0.118i)5-s + (0.398 − 0.0877i)6-s + (−0.262 − 0.0285i)7-s + (0.302 + 0.182i)8-s + (0.0891 + 0.321i)9-s + (−0.0928 − 0.0204i)10-s + (0.394 + 0.133i)11-s + (−0.135 + 0.255i)12-s + (0.283 − 1.01i)13-s + (0.135 − 0.128i)14-s + (0.0125 − 0.0765i)15-s + (−0.236 + 0.0798i)16-s + (1.17 − 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.913698 - 0.111447i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.913698 - 0.111447i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.647 - 0.762i)T \) |
| 3 | \( 1 + (0.796 + 0.605i)T \) |
| 59 | \( 1 + (6.83 + 3.50i)T \) |
| good | 5 | \( 1 + (-0.140 - 0.265i)T + (-2.80 + 4.13i)T^{2} \) |
| 7 | \( 1 + (0.694 + 0.0754i)T + (6.83 + 1.50i)T^{2} \) |
| 11 | \( 1 + (-1.30 - 0.441i)T + (8.75 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-1.02 + 3.67i)T + (-11.1 - 6.70i)T^{2} \) |
| 17 | \( 1 + (-4.84 + 0.527i)T + (16.6 - 3.65i)T^{2} \) |
| 19 | \( 1 + (-0.200 + 3.69i)T + (-18.8 - 2.05i)T^{2} \) |
| 23 | \( 1 + (-3.07 + 1.42i)T + (14.8 - 17.5i)T^{2} \) |
| 29 | \( 1 + (-2.55 - 3.01i)T + (-4.69 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-0.0121 - 0.223i)T + (-30.8 + 3.35i)T^{2} \) |
| 37 | \( 1 + (-3.78 + 2.27i)T + (17.3 - 32.6i)T^{2} \) |
| 41 | \( 1 + (-5.51 - 2.55i)T + (26.5 + 31.2i)T^{2} \) |
| 43 | \( 1 + (1.37 - 0.462i)T + (34.2 - 26.0i)T^{2} \) |
| 47 | \( 1 + (1.73 - 3.26i)T + (-26.3 - 38.9i)T^{2} \) |
| 53 | \( 1 + (-3.27 + 0.720i)T + (48.1 - 22.2i)T^{2} \) |
| 61 | \( 1 + (1.04 - 1.22i)T + (-9.86 - 60.1i)T^{2} \) |
| 67 | \( 1 + (1.39 + 0.842i)T + (31.3 + 59.1i)T^{2} \) |
| 71 | \( 1 + (3.17 - 5.99i)T + (-39.8 - 58.7i)T^{2} \) |
| 73 | \( 1 + (-3.29 + 3.12i)T + (3.95 - 72.8i)T^{2} \) |
| 79 | \( 1 + (8.61 - 6.54i)T + (21.1 - 76.1i)T^{2} \) |
| 83 | \( 1 + (2.75 + 6.92i)T + (-60.2 + 57.0i)T^{2} \) |
| 89 | \( 1 + (2.12 + 2.49i)T + (-14.3 + 87.8i)T^{2} \) |
| 97 | \( 1 + (-7.68 - 7.28i)T + (5.25 + 96.8i)T^{2} \) |
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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
−11.27098764838261814581555413799, −10.46232928609110424085172002808, −9.598150580032041310934223928738, −8.516859704697013959290757498008, −7.57749657908323079326913608668, −6.67679029551611918199361967013, −5.80678026526042967799246668163, −4.74423500721563259502525162397, −2.97304391420651026215631651236, −0.948073338216261801677324284072,
1.36577452965768156293867751506, 3.23012614468717475274707213847, 4.30773139262045183340534433858, 5.63841128549438336123929100273, 6.71663192311352237889200057170, 7.88468960061227689548219794733, 9.040978155265390536842128122030, 9.678784505131459607039544977907, 10.57596780926138533106522346574, 11.49438851696673797256955040673
