| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 3·9-s − 2·10-s + 2·13-s + 16-s − 6·17-s − 3·18-s + 4·19-s − 2·20-s + 8·23-s − 25-s + 2·26-s + 2·29-s − 31-s + 32-s − 6·34-s − 3·36-s + 10·37-s + 4·38-s − 2·40-s − 6·41-s + 8·43-s + 6·45-s + 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 9-s − 0.632·10-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.447·20-s + 1.66·23-s − 1/5·25-s + 0.392·26-s + 0.371·29-s − 0.179·31-s + 0.176·32-s − 1.02·34-s − 1/2·36-s + 1.64·37-s + 0.648·38-s − 0.316·40-s − 0.937·41-s + 1.21·43-s + 0.894·45-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.108740660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108740660\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.05691750776431447573088356666, −13.90986476156025600386975635688, −12.88268162615932246882266504036, −11.52935425193863190686584011034, −11.06274569152331522479839207162, −9.056498267458590117851162621111, −7.76857292337562945056886541715, −6.32858579420662158347624019611, −4.74203912193695697715499891246, −3.16188554299090604324575557253,
3.16188554299090604324575557253, 4.74203912193695697715499891246, 6.32858579420662158347624019611, 7.76857292337562945056886541715, 9.056498267458590117851162621111, 11.06274569152331522479839207162, 11.52935425193863190686584011034, 12.88268162615932246882266504036, 13.90986476156025600386975635688, 15.05691750776431447573088356666
