| L(s) = 1 | − 0.134·2-s + 3·3-s − 7.98·4-s − 0.404·6-s − 17.3·7-s + 2.15·8-s + 9·9-s + 42.1·11-s − 23.9·12-s − 84.5·13-s + 2.33·14-s + 63.5·16-s − 45.1·17-s − 1.21·18-s + 83.6·19-s − 51.9·21-s − 5.68·22-s + 20.9·23-s + 6.47·24-s + 11.4·26-s + 27·27-s + 138.·28-s + 220.·29-s + 160.·31-s − 25.8·32-s + 126.·33-s + 6.08·34-s + ⋯ |
| L(s) = 1 | − 0.0477·2-s + 0.577·3-s − 0.997·4-s − 0.0275·6-s − 0.934·7-s + 0.0953·8-s + 0.333·9-s + 1.15·11-s − 0.576·12-s − 1.80·13-s + 0.0445·14-s + 0.993·16-s − 0.643·17-s − 0.0159·18-s + 1.01·19-s − 0.539·21-s − 0.0551·22-s + 0.189·23-s + 0.0550·24-s + 0.0860·26-s + 0.192·27-s + 0.932·28-s + 1.40·29-s + 0.928·31-s − 0.142·32-s + 0.667·33-s + 0.0307·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.134T + 8T^{2} \) |
| 7 | \( 1 + 17.3T + 343T^{2} \) |
| 11 | \( 1 - 42.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 84.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 45.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 83.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 20.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 160.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 263.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 410.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 290.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 142.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 472.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 54.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 449.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 626.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 48.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 647.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 669.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 866.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 177.T + 9.12e5T^{2} \) |
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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
−8.676450499046043786702777717387, −7.74317837473344720913897501971, −6.98018639542787290649689041234, −6.18364765679119121053347822101, −4.95784809653597132994831468949, −4.38629757687279148957258203912, −3.38684435386933619179538889363, −2.61321557164591802991307964904, −1.13830233171167748025671801699, 0,
1.13830233171167748025671801699, 2.61321557164591802991307964904, 3.38684435386933619179538889363, 4.38629757687279148957258203912, 4.95784809653597132994831468949, 6.18364765679119121053347822101, 6.98018639542787290649689041234, 7.74317837473344720913897501971, 8.676450499046043786702777717387
