| L(s) = 1 | − 4.10·2-s + 3·3-s + 8.87·4-s − 17.1·5-s − 12.3·6-s − 4.94·7-s − 3.61·8-s + 9·9-s + 70.4·10-s − 3.96·11-s + 26.6·12-s + 87.8·13-s + 20.3·14-s − 51.4·15-s − 56.1·16-s − 129.·17-s − 36.9·18-s + 37.7·19-s − 152.·20-s − 14.8·21-s + 16.2·22-s − 10.8·24-s + 168.·25-s − 361.·26-s + 27·27-s − 43.9·28-s + 7.20·29-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 0.577·3-s + 1.10·4-s − 1.53·5-s − 0.838·6-s − 0.267·7-s − 0.159·8-s + 0.333·9-s + 2.22·10-s − 0.108·11-s + 0.640·12-s + 1.87·13-s + 0.388·14-s − 0.885·15-s − 0.877·16-s − 1.84·17-s − 0.484·18-s + 0.455·19-s − 1.70·20-s − 0.154·21-s + 0.157·22-s − 0.0922·24-s + 1.35·25-s − 2.72·26-s + 0.192·27-s − 0.296·28-s + 0.0461·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6286112134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6286112134\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 4.10T + 8T^{2} \) |
| 5 | \( 1 + 17.1T + 125T^{2} \) |
| 7 | \( 1 + 4.94T + 343T^{2} \) |
| 11 | \( 1 + 3.96T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 7.20T + 2.43e4T^{2} \) |
| 31 | \( 1 - 134.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 278.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 201.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 505.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 276.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 228.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 147.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 518.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 207.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 757.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 259.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 174.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 136.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.14e3T + 9.12e5T^{2} \) |
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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
−8.842548667066658322524408808473, −8.390984706901342935855393838072, −7.80153223136463856505364955721, −7.00570352507590956559711068260, −6.29092319620870334919138608469, −4.56079381207717773147320973032, −3.88673500995973002134932193987, −2.89148636731390881043642962703, −1.55172934809755235616672184031, −0.48367415202252198600685883877,
0.48367415202252198600685883877, 1.55172934809755235616672184031, 2.89148636731390881043642962703, 3.88673500995973002134932193987, 4.56079381207717773147320973032, 6.29092319620870334919138608469, 7.00570352507590956559711068260, 7.80153223136463856505364955721, 8.390984706901342935855393838072, 8.842548667066658322524408808473
