| L(s) = 1 | − 2·2-s − 5·3-s − 4·4-s − 6·5-s + 10·6-s − 8·7-s + 24·8-s − 2·9-s + 12·10-s + 34·11-s + 20·12-s − 57·13-s + 16·14-s + 30·15-s − 16·16-s − 80·17-s + 4·18-s − 70·19-s + 24·20-s + 40·21-s − 68·22-s + 23·23-s − 120·24-s − 89·25-s + 114·26-s + 145·27-s + 32·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.962·3-s − 1/2·4-s − 0.536·5-s + 0.680·6-s − 0.431·7-s + 1.06·8-s − 0.0740·9-s + 0.379·10-s + 0.931·11-s + 0.481·12-s − 1.21·13-s + 0.305·14-s + 0.516·15-s − 1/4·16-s − 1.14·17-s + 0.0523·18-s − 0.845·19-s + 0.268·20-s + 0.415·21-s − 0.658·22-s + 0.208·23-s − 1.02·24-s − 0.711·25-s + 0.859·26-s + 1.03·27-s + 0.215·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| good | 2 | \( 1 + p T + p^{3} T^{2} \) |
| 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 34 T + p^{3} T^{2} \) |
| 13 | \( 1 + 57 T + p^{3} T^{2} \) |
| 17 | \( 1 + 80 T + p^{3} T^{2} \) |
| 19 | \( 1 + 70 T + p^{3} T^{2} \) |
| 29 | \( 1 - 245 T + p^{3} T^{2} \) |
| 31 | \( 1 - 103 T + p^{3} T^{2} \) |
| 37 | \( 1 + 298 T + p^{3} T^{2} \) |
| 41 | \( 1 - 95 T + p^{3} T^{2} \) |
| 43 | \( 1 - 88 T + p^{3} T^{2} \) |
| 47 | \( 1 + 357 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 408 T + p^{3} T^{2} \) |
| 61 | \( 1 - 822 T + p^{3} T^{2} \) |
| 67 | \( 1 - 926 T + p^{3} T^{2} \) |
| 71 | \( 1 - 335 T + p^{3} T^{2} \) |
| 73 | \( 1 + 899 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1322 T + p^{3} T^{2} \) |
| 83 | \( 1 + 36 T + p^{3} T^{2} \) |
| 89 | \( 1 + 460 T + p^{3} T^{2} \) |
| 97 | \( 1 + 964 T + p^{3} T^{2} \) |
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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
−17.24025834356088088625422099288, −15.91568269315754517637383302708, −14.24824725063968658292713882395, −12.57363411656993674785053245336, −11.35826827100905508049444840313, −9.961523166511362664927358777418, −8.535712989536730575614351464131, −6.70152066336624718392450386020, −4.61063653410194406419918109269, 0,
4.61063653410194406419918109269, 6.70152066336624718392450386020, 8.535712989536730575614351464131, 9.961523166511362664927358777418, 11.35826827100905508049444840313, 12.57363411656993674785053245336, 14.24824725063968658292713882395, 15.91568269315754517637383302708, 17.24025834356088088625422099288
