| L(s) = 1 | − 1.28·3-s − 1.72·5-s − 0.336·7-s − 1.33·9-s − 13-s + 2.22·15-s − 4.46·17-s + 4.11·19-s + 0.433·21-s − 7.70·23-s − 2.01·25-s + 5.59·27-s − 6.71·29-s + 8.48·31-s + 0.580·35-s − 8.70·37-s + 1.28·39-s − 1.85·41-s − 10.9·43-s + 2.31·45-s + 7.05·47-s − 6.88·49-s + 5.75·51-s + 10.0·53-s − 5.30·57-s + 2.80·59-s + 5.58·61-s + ⋯ |
| L(s) = 1 | − 0.744·3-s − 0.772·5-s − 0.127·7-s − 0.446·9-s − 0.277·13-s + 0.574·15-s − 1.08·17-s + 0.943·19-s + 0.0945·21-s − 1.60·23-s − 0.403·25-s + 1.07·27-s − 1.24·29-s + 1.52·31-s + 0.0981·35-s − 1.43·37-s + 0.206·39-s − 0.289·41-s − 1.66·43-s + 0.344·45-s + 1.02·47-s − 0.983·49-s + 0.806·51-s + 1.37·53-s − 0.702·57-s + 0.365·59-s + 0.715·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3989034140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3989034140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.28T + 3T^{2} \) |
| 5 | \( 1 + 1.72T + 5T^{2} \) |
| 7 | \( 1 + 0.336T + 7T^{2} \) |
| 17 | \( 1 + 4.46T + 17T^{2} \) |
| 19 | \( 1 - 4.11T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 + 1.85T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 7.05T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 2.80T + 59T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 2.08T + 71T^{2} \) |
| 73 | \( 1 - 7.20T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 9.30T + 97T^{2} \) |
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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
−8.136201604403004314012835784891, −7.21485919485343606654277799644, −6.67297817357433626422492154814, −5.83487636654545574160612580596, −5.28361144628611061314903505613, −4.41597464119288521935494713423, −3.74207004912456519745335769667, −2.83349916022880498716782553588, −1.77469084014229645563041728256, −0.32903282393543310065116367560,
0.32903282393543310065116367560, 1.77469084014229645563041728256, 2.83349916022880498716782553588, 3.74207004912456519745335769667, 4.41597464119288521935494713423, 5.28361144628611061314903505613, 5.83487636654545574160612580596, 6.67297817357433626422492154814, 7.21485919485343606654277799644, 8.136201604403004314012835784891
