| L(s) = 1 | − 354.·2-s + 1.65e4·3-s − 5.40e3·4-s − 1.34e6·5-s − 5.86e6·6-s + 4.33e6·7-s + 4.83e7·8-s + 1.44e8·9-s + 4.75e8·10-s + 1.76e8·11-s − 8.95e7·12-s + 1.02e9·13-s − 1.53e9·14-s − 2.22e10·15-s − 1.64e10·16-s + 2.73e10·17-s − 5.13e10·18-s − 4.65e10·19-s + 7.26e9·20-s + 7.18e10·21-s − 6.24e10·22-s − 7.83e10·23-s + 8.00e11·24-s + 1.03e12·25-s − 3.61e11·26-s + 2.59e11·27-s − 2.34e10·28-s + ⋯ |
| L(s) = 1 | − 0.979·2-s + 1.45·3-s − 0.0412·4-s − 1.53·5-s − 1.42·6-s + 0.284·7-s + 1.01·8-s + 1.12·9-s + 1.50·10-s + 0.247·11-s − 0.0601·12-s + 0.346·13-s − 0.278·14-s − 2.23·15-s − 0.957·16-s + 0.951·17-s − 1.09·18-s − 0.629·19-s + 0.0634·20-s + 0.414·21-s − 0.242·22-s − 0.208·23-s + 1.48·24-s + 1.36·25-s − 0.339·26-s + 0.176·27-s − 0.0117·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 7.83e10T \) |
| good | 2 | \( 1 + 354.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.65e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 1.34e6T + 7.62e11T^{2} \) |
| 7 | \( 1 - 4.33e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.76e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 1.02e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 2.73e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 4.65e10T + 5.48e21T^{2} \) |
| 29 | \( 1 + 2.50e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 7.58e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.28e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 6.57e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 7.27e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.98e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.45e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.35e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.35e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 4.65e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 8.91e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 3.32e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 8.11e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 7.11e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 2.05e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 9.58e16T + 5.95e33T^{2} \) |
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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
−13.49754222932862288060947643697, −11.84651098626936318603738256195, −10.29061464199915995613066470551, −8.847568521788251031292483967449, −8.190343753165733411118330723254, −7.38508106099475799884168836292, −4.35420476214959749271432847890, −3.31007298852646072528460507960, −1.49826740788039819008212072239, 0,
1.49826740788039819008212072239, 3.31007298852646072528460507960, 4.35420476214959749271432847890, 7.38508106099475799884168836292, 8.190343753165733411118330723254, 8.847568521788251031292483967449, 10.29061464199915995613066470551, 11.84651098626936318603738256195, 13.49754222932862288060947643697
