| L(s) = 1 | − 5-s + 3.15·7-s − 0.556·11-s − 5.71·13-s − 17-s + 3.15·19-s + 7.71·23-s + 25-s − 6.55·29-s + 9.71·31-s − 3.15·35-s + 0.844·37-s − 7.75·41-s + 10.3·43-s − 8.86·47-s + 2.95·49-s + 8.86·53-s + 0.556·55-s + 10.0·59-s + 8.59·61-s + 5.71·65-s + 0.288·67-s − 0.267·73-s − 1.75·77-s + 3.11·79-s + 8.31·83-s + 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.19·7-s − 0.167·11-s − 1.58·13-s − 0.242·17-s + 0.723·19-s + 1.60·23-s + 0.200·25-s − 1.21·29-s + 1.74·31-s − 0.533·35-s + 0.138·37-s − 1.21·41-s + 1.57·43-s − 1.29·47-s + 0.422·49-s + 1.21·53-s + 0.0749·55-s + 1.30·59-s + 1.10·61-s + 0.708·65-s + 0.0352·67-s − 0.0313·73-s − 0.199·77-s + 0.350·79-s + 0.912·83-s + 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.810561124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.810561124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 + 0.556T + 11T^{2} \) |
| 13 | \( 1 + 5.71T + 13T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 - 7.71T + 23T^{2} \) |
| 29 | \( 1 + 6.55T + 29T^{2} \) |
| 31 | \( 1 - 9.71T + 31T^{2} \) |
| 37 | \( 1 - 0.844T + 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 8.86T + 47T^{2} \) |
| 53 | \( 1 - 8.86T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 8.59T + 61T^{2} \) |
| 67 | \( 1 - 0.288T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 0.267T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 - 8.31T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 9.11T + 97T^{2} \) |
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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
−8.617798222583312168488729559964, −7.86907730654284462824370066572, −7.35626394883629483750948282091, −6.65723389100345516709436480496, −5.24578365996980434649679414346, −5.03107312686932211727182512900, −4.13580753480268436972001059935, −2.98464413257120657654837827304, −2.11355010830326135941874146329, −0.827080362537202125897739455347,
0.827080362537202125897739455347, 2.11355010830326135941874146329, 2.98464413257120657654837827304, 4.13580753480268436972001059935, 5.03107312686932211727182512900, 5.24578365996980434649679414346, 6.65723389100345516709436480496, 7.35626394883629483750948282091, 7.86907730654284462824370066572, 8.617798222583312168488729559964
