| L(s) = 1 | − 553.·2-s − 346.·3-s + 1.75e5·4-s + 6.41e5·5-s + 1.91e5·6-s − 1.77e7·7-s − 2.43e7·8-s − 1.29e8·9-s − 3.55e8·10-s + 1.27e9·11-s − 6.06e7·12-s + 2.71e9·13-s + 9.79e9·14-s − 2.22e8·15-s − 9.46e9·16-s + 1.35e10·17-s + 7.13e10·18-s − 6.90e10·19-s + 1.12e11·20-s + 6.12e9·21-s − 7.05e11·22-s + 4.68e11·23-s + 8.43e9·24-s − 3.50e11·25-s − 1.50e12·26-s + 8.93e10·27-s − 3.10e12·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 0.0304·3-s + 1.33·4-s + 0.734·5-s + 0.0465·6-s − 1.16·7-s − 0.513·8-s − 0.999·9-s − 1.12·10-s + 1.79·11-s − 0.0406·12-s + 0.922·13-s + 1.77·14-s − 0.0223·15-s − 0.551·16-s + 0.472·17-s + 1.52·18-s − 0.932·19-s + 0.981·20-s + 0.0353·21-s − 2.74·22-s + 1.24·23-s + 0.0156·24-s − 0.459·25-s − 1.41·26-s + 0.0608·27-s − 1.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.8597411167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8597411167\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 5.00e11T \) |
| good | 2 | \( 1 + 553.T + 1.31e5T^{2} \) |
| 3 | \( 1 + 346.T + 1.29e8T^{2} \) |
| 5 | \( 1 - 6.41e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.77e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.27e9T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.71e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.35e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 6.90e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 4.68e11T + 1.41e23T^{2} \) |
| 31 | \( 1 + 7.33e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.41e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 3.62e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.17e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + 2.50e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 8.36e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 6.11e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.58e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 1.78e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 4.49e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.35e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.86e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.42e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 6.42e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 5.31e16T + 5.95e33T^{2} \) |
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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
−13.27265021236108007210353107580, −11.63382993548036061922641858405, −10.44191629746640287317701362809, −9.220610560113174944670899293655, −8.778476632149064319600718515355, −6.87747441803954297259827039578, −5.99783404568191868576898528720, −3.49021725401879573421041174589, −1.85103311738499055021103935940, −0.65550265170396099872496259602,
0.65550265170396099872496259602, 1.85103311738499055021103935940, 3.49021725401879573421041174589, 5.99783404568191868576898528720, 6.87747441803954297259827039578, 8.778476632149064319600718515355, 9.220610560113174944670899293655, 10.44191629746640287317701362809, 11.63382993548036061922641858405, 13.27265021236108007210353107580
