| L(s) = 1 | − 2.51·2-s + 0.871·3-s + 4.34·4-s + 3.48·5-s − 2.19·6-s − 2.34·7-s − 5.90·8-s − 2.24·9-s − 8.78·10-s + 3.35·11-s + 3.78·12-s + 5.89·14-s + 3.03·15-s + 6.19·16-s − 6.40·17-s + 5.64·18-s + 2.55·19-s + 15.1·20-s − 2.04·21-s − 8.45·22-s − 1.79·23-s − 5.15·24-s + 7.15·25-s − 4.56·27-s − 10.1·28-s + 8.57·29-s − 7.65·30-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.503·3-s + 2.17·4-s + 1.55·5-s − 0.896·6-s − 0.884·7-s − 2.08·8-s − 0.746·9-s − 2.77·10-s + 1.01·11-s + 1.09·12-s + 1.57·14-s + 0.784·15-s + 1.54·16-s − 1.55·17-s + 1.32·18-s + 0.585·19-s + 3.38·20-s − 0.445·21-s − 1.80·22-s − 0.375·23-s − 1.05·24-s + 1.43·25-s − 0.879·27-s − 1.92·28-s + 1.59·29-s − 1.39·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.144055606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.144055606\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 - 0.871T + 3T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 17 | \( 1 + 6.40T + 17T^{2} \) |
| 19 | \( 1 - 2.55T + 19T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 - 8.57T + 29T^{2} \) |
| 37 | \( 1 - 8.40T + 37T^{2} \) |
| 41 | \( 1 - 5.95T + 41T^{2} \) |
| 43 | \( 1 - 3.26T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 + 9.35T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 3.59T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 + 5.97T + 73T^{2} \) |
| 79 | \( 1 - 1.26T + 79T^{2} \) |
| 83 | \( 1 - 7.88T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 0.0508T + 97T^{2} \) |
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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
−8.540359450120649974266745494653, −7.68887761078197471656465611711, −6.76016677135174999765321386243, −6.27744789740846081141440576476, −5.90598785555273057491129641022, −4.49845181868217261637871560288, −3.10579500412178558316623119267, −2.49811547335315966637540413482, −1.80761371893281088390284054447, −0.73036991840818847577178050440,
0.73036991840818847577178050440, 1.80761371893281088390284054447, 2.49811547335315966637540413482, 3.10579500412178558316623119267, 4.49845181868217261637871560288, 5.90598785555273057491129641022, 6.27744789740846081141440576476, 6.76016677135174999765321386243, 7.68887761078197471656465611711, 8.540359450120649974266745494653
