| L(s) = 1 | − 2.35·3-s + 1.14·7-s + 2.53·9-s − 2.32·11-s + 3.91·13-s − 3.47·17-s − 6.32·19-s − 2.70·21-s + 23-s + 1.09·27-s + 5.03·29-s + 1.11·31-s + 5.47·33-s − 3.00·37-s − 9.20·39-s + 1.50·41-s + 5.03·43-s − 2.76·47-s − 5.67·49-s + 8.18·51-s + 9.41·53-s + 14.8·57-s − 6.90·59-s + 3.95·61-s + 2.91·63-s + 0.705·67-s − 2.35·69-s + ⋯ |
| L(s) = 1 | − 1.35·3-s + 0.434·7-s + 0.845·9-s − 0.702·11-s + 1.08·13-s − 0.843·17-s − 1.45·19-s − 0.590·21-s + 0.208·23-s + 0.210·27-s + 0.935·29-s + 0.200·31-s + 0.953·33-s − 0.494·37-s − 1.47·39-s + 0.234·41-s + 0.767·43-s − 0.403·47-s − 0.811·49-s + 1.14·51-s + 1.29·53-s + 1.97·57-s − 0.898·59-s + 0.505·61-s + 0.367·63-s + 0.0862·67-s − 0.283·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8947259484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8947259484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 + 2.32T + 11T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 17 | \( 1 + 3.47T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 29 | \( 1 - 5.03T + 29T^{2} \) |
| 31 | \( 1 - 1.11T + 31T^{2} \) |
| 37 | \( 1 + 3.00T + 37T^{2} \) |
| 41 | \( 1 - 1.50T + 41T^{2} \) |
| 43 | \( 1 - 5.03T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 + 6.90T + 59T^{2} \) |
| 61 | \( 1 - 3.95T + 61T^{2} \) |
| 67 | \( 1 - 0.705T + 67T^{2} \) |
| 71 | \( 1 + 7.42T + 71T^{2} \) |
| 73 | \( 1 - 2.08T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 - 7.38T + 83T^{2} \) |
| 89 | \( 1 + 0.472T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−8.492952448935303423248787695464, −7.48414988823974201122673160706, −6.57721247827333608139812606880, −6.19402043677978040331750318678, −5.42649958662313118626661062000, −4.68452290340300597096281646398, −4.10770783067695857411551166022, −2.82751524799688973680976357093, −1.73739775869836256276129027999, −0.56595207839577096277145334085,
0.56595207839577096277145334085, 1.73739775869836256276129027999, 2.82751524799688973680976357093, 4.10770783067695857411551166022, 4.68452290340300597096281646398, 5.42649958662313118626661062000, 6.19402043677978040331750318678, 6.57721247827333608139812606880, 7.48414988823974201122673160706, 8.492952448935303423248787695464
