L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s + 4·11-s − 12-s − 2·13-s + 2·15-s + 16-s − 6·17-s + 18-s + 19-s − 2·20-s + 4·22-s + 4·23-s − 24-s − 25-s − 2·26-s − 27-s + 2·29-s + 2·30-s − 4·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s + 0.852·22-s + 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 251826 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 251826 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.322630075\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.322630075\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.74297901507170, −12.43462937457507, −11.81391360100877, −11.43286461235290, −11.26580564080391, −10.90322061376708, −10.02436782651476, −9.688716905000626, −9.218012431220391, −8.453551342931704, −8.221307061492515, −7.475454729880147, −7.017571904862577, −6.515714710178007, −6.463648245371707, −5.551572353714202, −5.013765997710602, −4.679378671983044, −4.118477272318379, −3.673018020796143, −3.245265782858974, −2.351721662242580, −1.913280431174343, −1.068823099277007, −0.4201927012218417,
0.4201927012218417, 1.068823099277007, 1.913280431174343, 2.351721662242580, 3.245265782858974, 3.673018020796143, 4.118477272318379, 4.679378671983044, 5.013765997710602, 5.551572353714202, 6.463648245371707, 6.515714710178007, 7.017571904862577, 7.475454729880147, 8.221307061492515, 8.453551342931704, 9.218012431220391, 9.688716905000626, 10.02436782651476, 10.90322061376708, 11.26580564080391, 11.43286461235290, 11.81391360100877, 12.43462937457507, 12.74297901507170