| L(s) = 1 | + (0.289 + 0.105i)2-s + (−0.285 − 1.61i)3-s + (−1.45 − 1.22i)4-s + (0.766 − 0.642i)5-s + (0.0879 − 0.498i)6-s + (−0.0445 + 0.0772i)7-s + (−0.601 − 1.04i)8-s + (0.279 − 0.101i)9-s + (0.289 − 0.105i)10-s + (1.68 + 2.91i)11-s + (−1.56 + 2.71i)12-s + (0.0369 − 0.209i)13-s + (−0.0210 + 0.0176i)14-s + (−1.25 − 1.05i)15-s + (0.597 + 3.38i)16-s + (2.36 + 0.859i)17-s + ⋯ |
| L(s) = 1 | + (0.204 + 0.0744i)2-s + (−0.164 − 0.934i)3-s + (−0.729 − 0.612i)4-s + (0.342 − 0.287i)5-s + (0.0358 − 0.203i)6-s + (−0.0168 + 0.0291i)7-s + (−0.212 − 0.368i)8-s + (0.0931 − 0.0339i)9-s + (0.0915 − 0.0333i)10-s + (0.507 + 0.879i)11-s + (−0.452 + 0.782i)12-s + (0.0102 − 0.0581i)13-s + (−0.00562 + 0.00471i)14-s + (−0.325 − 0.272i)15-s + (0.149 + 0.846i)16-s + (0.573 + 0.208i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.801578 - 0.557626i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.801578 - 0.557626i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-0.949 - 4.25i)T \) |
| good | 2 | \( 1 + (-0.289 - 0.105i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (0.285 + 1.61i)T + (-2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (0.0445 - 0.0772i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.68 - 2.91i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0369 + 0.209i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.36 - 0.859i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (4.57 + 3.83i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.51 + 1.64i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (4.03 - 6.98i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 + (0.523 + 2.96i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.87 - 1.57i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.15 + 2.60i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (6.43 + 5.39i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (9.80 + 3.56i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.757 + 0.635i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (9.37 - 3.41i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.73 + 3.97i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.73 - 15.5i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.178 + 1.01i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (8.96 - 15.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.113 + 0.646i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (15.7 + 5.75i)T + (74.3 + 62.3i)T^{2} \) |
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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
−13.85938792235623357978116537883, −12.57176963214683648029068061529, −12.28201971828227927970057113013, −10.34963048497920775003470027735, −9.521667427616390307140170519264, −8.167539645474115529240445937376, −6.75436420886553543660391315704, −5.70261461665257519359825228431, −4.24795213040649648476181885417, −1.50458090544965156060350167378,
3.31351255353067939222715221474, 4.48886889489644814220878016055, 5.77949338351333886110231164885, 7.56842323178309201110647357839, 8.995530121615740573835254547102, 9.760119043870399524910614290020, 10.98498747871912552023347118810, 12.02735158004872362257507064715, 13.38083821824791183554619900994, 14.02880802861937871891786709567
