| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 10.3·5-s − 6·6-s − 35.0·7-s + 8·8-s + 9·9-s + 20.7·10-s + 0.279·11-s − 12·12-s + 74.4·13-s − 70.0·14-s − 31.1·15-s + 16·16-s − 78.9·17-s + 18·18-s + 41.5·20-s + 105.·21-s + 0.558·22-s + 45.1·23-s − 24·24-s − 17.1·25-s + 148.·26-s − 27·27-s − 140.·28-s + 263.·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.928·5-s − 0.408·6-s − 1.89·7-s + 0.353·8-s + 0.333·9-s + 0.656·10-s + 0.00764·11-s − 0.288·12-s + 1.58·13-s − 1.33·14-s − 0.536·15-s + 0.250·16-s − 1.12·17-s + 0.235·18-s + 0.464·20-s + 1.09·21-s + 0.00540·22-s + 0.409·23-s − 0.204·24-s − 0.137·25-s + 1.12·26-s − 0.192·27-s − 0.945·28-s + 1.68·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 10.3T + 125T^{2} \) |
| 7 | \( 1 + 35.0T + 343T^{2} \) |
| 11 | \( 1 - 0.279T + 1.33e3T^{2} \) |
| 13 | \( 1 - 74.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.9T + 4.91e3T^{2} \) |
| 23 | \( 1 - 45.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 263.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 307.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 163.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 340.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 681.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 255.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 113.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 969.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 42.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 598.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 372.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 638.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 211.T + 9.12e5T^{2} \) |
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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
−8.512335133989899558577118460142, −6.88871854240405944261073174458, −6.67752917535077964611383120235, −5.94530402240290793754907447812, −5.42412598103433470840163433329, −4.17159101596236023513230620902, −3.43822793505415825157507634602, −2.51371763622011327096527382105, −1.32690158173488221690359438855, 0,
1.32690158173488221690359438855, 2.51371763622011327096527382105, 3.43822793505415825157507634602, 4.17159101596236023513230620902, 5.42412598103433470840163433329, 5.94530402240290793754907447812, 6.67752917535077964611383120235, 6.88871854240405944261073174458, 8.512335133989899558577118460142
