| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 19.0·5-s − 6·6-s − 33.7·7-s − 8·8-s + 9·9-s + 38.0·10-s + 59.8·11-s + 12·12-s + 67.5·14-s − 57.0·15-s + 16·16-s + 37.3·17-s − 18·18-s + 11.1·19-s − 76.1·20-s − 101.·21-s − 119.·22-s + 51.6·23-s − 24·24-s + 237.·25-s + 27·27-s − 135.·28-s − 48.0·29-s + 114.·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.70·5-s − 0.408·6-s − 1.82·7-s − 0.353·8-s + 0.333·9-s + 1.20·10-s + 1.64·11-s + 0.288·12-s + 1.28·14-s − 0.982·15-s + 0.250·16-s + 0.532·17-s − 0.235·18-s + 0.134·19-s − 0.851·20-s − 1.05·21-s − 1.16·22-s + 0.467·23-s − 0.204·24-s + 1.89·25-s + 0.192·27-s − 0.911·28-s − 0.307·29-s + 0.694·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 19.0T + 125T^{2} \) |
| 7 | \( 1 + 33.7T + 343T^{2} \) |
| 11 | \( 1 - 59.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 37.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 51.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 48.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 66.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 224.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 9.76T + 7.95e4T^{2} \) |
| 47 | \( 1 - 119.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 466.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 721.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 597.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 364.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 267.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 801.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 931.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 179.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 648.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 803.T + 9.12e5T^{2} \) |
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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
−9.260594630950993935098317016258, −8.394611377486829213829924168439, −7.52457525892753655830339171864, −6.86919445504842798126446932534, −6.17065003134345715541465300762, −4.32377924464597119811316249328, −3.52420838346966044240996851495, −2.98870868956623204387325437471, −1.10291089264514530704732930720, 0,
1.10291089264514530704732930720, 2.98870868956623204387325437471, 3.52420838346966044240996851495, 4.32377924464597119811316249328, 6.17065003134345715541465300762, 6.86919445504842798126446932534, 7.52457525892753655830339171864, 8.394611377486829213829924168439, 9.260594630950993935098317016258
