| L(s) = 1 | + 3.16·5-s − 1.74·7-s − 6.47·11-s + 1.41·13-s − 2.47·17-s − 6.47·19-s + 5.65·23-s + 5.00·25-s + 5.99·29-s + 3.90·31-s − 5.52·35-s + 10.5·37-s + 2.47·41-s + 1.52·43-s − 3.94·49-s − 11.6·53-s − 20.4·55-s − 8.94·59-s + 2.08·61-s + 4.47·65-s + 12·67-s − 9.15·71-s + 2.94·73-s + 11.3·77-s + 7.40·79-s + 6.47·83-s − 7.81·85-s + ⋯ |
| L(s) = 1 | + 1.41·5-s − 0.660·7-s − 1.95·11-s + 0.392·13-s − 0.599·17-s − 1.48·19-s + 1.17·23-s + 1.00·25-s + 1.11·29-s + 0.702·31-s − 0.934·35-s + 1.73·37-s + 0.386·41-s + 0.232·43-s − 0.563·49-s − 1.59·53-s − 2.75·55-s − 1.16·59-s + 0.266·61-s + 0.554·65-s + 1.46·67-s − 1.08·71-s + 0.344·73-s + 1.28·77-s + 0.833·79-s + 0.710·83-s − 0.847·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.972617060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.972617060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 + 6.47T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 31 | \( 1 - 3.90T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 2.08T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 9.15T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 - 7.40T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.84956601433291396327904072846, −6.77781592904982056460552693081, −6.29853040028054884943853952762, −5.82349657576651864873492994742, −4.93740213913496464232968464666, −4.48822459196255333159797730374, −3.09488433174261509832197533352, −2.61920292831551753897990457283, −1.94836447212783504927597628299, −0.64718828989727598030993658141,
0.64718828989727598030993658141, 1.94836447212783504927597628299, 2.61920292831551753897990457283, 3.09488433174261509832197533352, 4.48822459196255333159797730374, 4.93740213913496464232968464666, 5.82349657576651864873492994742, 6.29853040028054884943853952762, 6.77781592904982056460552693081, 7.84956601433291396327904072846
