| L(s) = 1 | + 5-s + 3·11-s − 6·13-s + 5·17-s − 19-s − 7·23-s − 4·25-s − 2·29-s + 5·31-s + 3·37-s + 2·41-s + 4·43-s + 5·47-s + 53-s + 3·55-s + 15·59-s − 5·61-s − 6·65-s + 9·67-s + 7·73-s − 79-s + 12·83-s + 5·85-s − 7·89-s − 95-s − 2·97-s − 3·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.904·11-s − 1.66·13-s + 1.21·17-s − 0.229·19-s − 1.45·23-s − 4/5·25-s − 0.371·29-s + 0.898·31-s + 0.493·37-s + 0.312·41-s + 0.609·43-s + 0.729·47-s + 0.137·53-s + 0.404·55-s + 1.95·59-s − 0.640·61-s − 0.744·65-s + 1.09·67-s + 0.819·73-s − 0.112·79-s + 1.31·83-s + 0.542·85-s − 0.741·89-s − 0.102·95-s − 0.203·97-s − 0.298·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.056012954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.056012954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.80938945161247491825763012745, −7.35985502139519586381952817992, −6.46749185565806654852829644076, −5.85480315263226514984617937653, −5.19716917776365216970698814465, −4.30031124746519077999263005678, −3.66938576158705372020834388933, −2.54985724314154952520724050951, −1.93327733778023307163773954158, −0.72268694847289270306452538024,
0.72268694847289270306452538024, 1.93327733778023307163773954158, 2.54985724314154952520724050951, 3.66938576158705372020834388933, 4.30031124746519077999263005678, 5.19716917776365216970698814465, 5.85480315263226514984617937653, 6.46749185565806654852829644076, 7.35985502139519586381952817992, 7.80938945161247491825763012745
