| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 2·11-s + 7·13-s − 14-s + 16-s + 7·17-s + 8·19-s − 2·22-s + 5·23-s − 7·26-s + 28-s − 9·29-s + 31-s − 32-s − 7·34-s − 2·37-s − 8·38-s − 11·41-s + 3·43-s + 2·44-s − 5·46-s − 4·47-s + 49-s + 7·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.603·11-s + 1.94·13-s − 0.267·14-s + 1/4·16-s + 1.69·17-s + 1.83·19-s − 0.426·22-s + 1.04·23-s − 1.37·26-s + 0.188·28-s − 1.67·29-s + 0.179·31-s − 0.176·32-s − 1.20·34-s − 0.328·37-s − 1.29·38-s − 1.71·41-s + 0.457·43-s + 0.301·44-s − 0.737·46-s − 0.583·47-s + 1/7·49-s + 0.970·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.853888543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.853888543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.821648821316085235113260169228, −7.86087439724227841038035085522, −7.46903630537801045214345517800, −6.47201154785901138794964613757, −5.73443884214178855015116483866, −5.03532957760104748285298394108, −3.52399453372567038953215375137, −3.32354251010430335657934626486, −1.57259310028887997451852526697, −1.06785029157502222685241757336,
1.06785029157502222685241757336, 1.57259310028887997451852526697, 3.32354251010430335657934626486, 3.52399453372567038953215375137, 5.03532957760104748285298394108, 5.73443884214178855015116483866, 6.47201154785901138794964613757, 7.46903630537801045214345517800, 7.86087439724227841038035085522, 8.821648821316085235113260169228
