| L(s) = 1 | − 2·5-s + 7-s − 2·13-s + 4·17-s + 4·19-s + 6·23-s − 25-s + 2·29-s − 2·31-s − 2·35-s + 2·37-s + 6·43-s + 12·47-s + 49-s + 12·59-s + 2·61-s + 4·65-s + 4·67-s + 6·71-s − 10·73-s + 16·79-s + 6·83-s − 8·85-s + 6·89-s − 2·91-s − 8·95-s − 6·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 0.554·13-s + 0.970·17-s + 0.917·19-s + 1.25·23-s − 1/5·25-s + 0.371·29-s − 0.359·31-s − 0.338·35-s + 0.328·37-s + 0.914·43-s + 1.75·47-s + 1/7·49-s + 1.56·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.712·71-s − 1.17·73-s + 1.80·79-s + 0.658·83-s − 0.867·85-s + 0.635·89-s − 0.209·91-s − 0.820·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.507987110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.507987110\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.36688614091338, −13.85417065446383, −13.30726445661809, −12.65068885701993, −12.12934742537017, −11.91286611739166, −11.22408956607150, −10.89042219682029, −10.22527677757610, −9.659192395809082, −9.173688380214732, −8.558503058594676, −7.992764901079372, −7.379135878058741, −7.330775646137443, −6.491140272730176, −5.655688064280380, −5.267382409784044, −4.685982838471223, −3.972057352673995, −3.515973873602858, −2.805675112796777, −2.175969381511351, −1.094153696627846, −0.6436710329112358,
0.6436710329112358, 1.094153696627846, 2.175969381511351, 2.805675112796777, 3.515973873602858, 3.972057352673995, 4.685982838471223, 5.267382409784044, 5.655688064280380, 6.491140272730176, 7.330775646137443, 7.379135878058741, 7.992764901079372, 8.558503058594676, 9.173688380214732, 9.659192395809082, 10.22527677757610, 10.89042219682029, 11.22408956607150, 11.91286611739166, 12.12934742537017, 12.65068885701993, 13.30726445661809, 13.85417065446383, 14.36688614091338
