| L(s) = 1 | + 3.60·2-s + 1.17·3-s + 4.98·4-s − 5.88·5-s + 4.25·6-s + 15.2·7-s − 10.8·8-s − 25.6·9-s − 21.1·10-s + 45.5·11-s + 5.88·12-s − 26.6·13-s + 54.8·14-s − 6.93·15-s − 79.0·16-s − 95.3·17-s − 92.2·18-s − 111.·19-s − 29.3·20-s + 17.9·21-s + 164.·22-s + 23·23-s − 12.8·24-s − 90.4·25-s − 96.1·26-s − 62.0·27-s + 75.9·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.226·3-s + 0.623·4-s − 0.526·5-s + 0.289·6-s + 0.822·7-s − 0.479·8-s − 0.948·9-s − 0.670·10-s + 1.24·11-s + 0.141·12-s − 0.569·13-s + 1.04·14-s − 0.119·15-s − 1.23·16-s − 1.36·17-s − 1.20·18-s − 1.34·19-s − 0.327·20-s + 0.186·21-s + 1.59·22-s + 0.208·23-s − 0.108·24-s − 0.723·25-s − 0.725·26-s − 0.442·27-s + 0.512·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 3.60T + 8T^{2} \) |
| 3 | \( 1 - 1.17T + 27T^{2} \) |
| 5 | \( 1 + 5.88T + 125T^{2} \) |
| 7 | \( 1 - 15.2T + 343T^{2} \) |
| 11 | \( 1 - 45.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 111.T + 6.85e3T^{2} \) |
| 31 | \( 1 - 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 306.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 163.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 391.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 304.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 636.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 213.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 713.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 782.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 988.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 438.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 278.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 726.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 469.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.51T + 9.12e5T^{2} \) |
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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
−9.543776022505578301400628865021, −8.692645749524687704732752162045, −8.015701441802829299365350398302, −6.63627340691083436404960055876, −6.05101391555987096326850785305, −4.65168776780159106121938591305, −4.35080377585684307703707771098, −3.15158923846252019329407414996, −2.03581196259872966505153237334, 0,
2.03581196259872966505153237334, 3.15158923846252019329407414996, 4.35080377585684307703707771098, 4.65168776780159106121938591305, 6.05101391555987096326850785305, 6.63627340691083436404960055876, 8.015701441802829299365350398302, 8.692645749524687704732752162045, 9.543776022505578301400628865021
