| L(s) = 1 | − 2.48·3-s − 1.86·5-s − 0.813·7-s + 3.18·9-s − 4.32·11-s + 4.51·13-s + 4.62·15-s − 2.28·17-s + 19-s + 2.02·21-s + 8.58·23-s − 1.53·25-s − 0.471·27-s + 1.83·29-s − 7.94·31-s + 10.7·33-s + 1.51·35-s − 2.25·37-s − 11.2·39-s − 2.05·41-s − 9.48·43-s − 5.93·45-s + 1.54·47-s − 6.33·49-s + 5.67·51-s − 7.20·53-s + 8.04·55-s + ⋯ |
| L(s) = 1 | − 1.43·3-s − 0.831·5-s − 0.307·7-s + 1.06·9-s − 1.30·11-s + 1.25·13-s + 1.19·15-s − 0.553·17-s + 0.229·19-s + 0.441·21-s + 1.79·23-s − 0.307·25-s − 0.0907·27-s + 0.341·29-s − 1.42·31-s + 1.87·33-s + 0.255·35-s − 0.370·37-s − 1.79·39-s − 0.321·41-s − 1.44·43-s − 0.884·45-s + 0.224·47-s − 0.905·49-s + 0.795·51-s − 0.989·53-s + 1.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4045686942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4045686942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 2.48T + 3T^{2} \) |
| 5 | \( 1 + 1.86T + 5T^{2} \) |
| 7 | \( 1 + 0.813T + 7T^{2} \) |
| 11 | \( 1 + 4.32T + 11T^{2} \) |
| 13 | \( 1 - 4.51T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 + 7.94T + 31T^{2} \) |
| 37 | \( 1 + 2.25T + 37T^{2} \) |
| 41 | \( 1 + 2.05T + 41T^{2} \) |
| 43 | \( 1 + 9.48T + 43T^{2} \) |
| 47 | \( 1 - 1.54T + 47T^{2} \) |
| 53 | \( 1 + 7.20T + 53T^{2} \) |
| 59 | \( 1 + 7.28T + 59T^{2} \) |
| 61 | \( 1 + 1.64T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 + 8.14T + 71T^{2} \) |
| 73 | \( 1 - 6.16T + 73T^{2} \) |
| 83 | \( 1 + 6.53T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 6.50T + 97T^{2} \) |
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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
−7.991763495494060080473306862780, −7.22749885834852463543571484104, −6.61058540090117323638987397687, −5.93319739873617794348742000969, −5.15459781162798838758426017058, −4.74500506916230909948201421294, −3.66560120628784148846203176039, −2.99967631783385674437306191038, −1.54085907191368527484188581364, −0.36830981330242239243356222795,
0.36830981330242239243356222795, 1.54085907191368527484188581364, 2.99967631783385674437306191038, 3.66560120628784148846203176039, 4.74500506916230909948201421294, 5.15459781162798838758426017058, 5.93319739873617794348742000969, 6.61058540090117323638987397687, 7.22749885834852463543571484104, 7.991763495494060080473306862780
