| L(s) = 1 | − 1.94·2-s − 3·3-s − 4.20·4-s + 15.7·5-s + 5.84·6-s + 3.79·7-s + 23.7·8-s + 9·9-s − 30.6·10-s + 11·11-s + 12.6·12-s + 55.4·13-s − 7.40·14-s − 47.2·15-s − 12.6·16-s + 20.7·17-s − 17.5·18-s − 32.9·19-s − 66.2·20-s − 11.3·21-s − 21.4·22-s − 38.9·23-s − 71.3·24-s + 123.·25-s − 107.·26-s − 27·27-s − 15.9·28-s + ⋯ |
| L(s) = 1 | − 0.688·2-s − 0.577·3-s − 0.525·4-s + 1.40·5-s + 0.397·6-s + 0.205·7-s + 1.05·8-s + 0.333·9-s − 0.970·10-s + 0.301·11-s + 0.303·12-s + 1.18·13-s − 0.141·14-s − 0.813·15-s − 0.198·16-s + 0.296·17-s − 0.229·18-s − 0.398·19-s − 0.740·20-s − 0.118·21-s − 0.207·22-s − 0.352·23-s − 0.606·24-s + 0.985·25-s − 0.814·26-s − 0.192·27-s − 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 1.94T + 8T^{2} \) |
| 5 | \( 1 - 15.7T + 125T^{2} \) |
| 7 | \( 1 - 3.79T + 343T^{2} \) |
| 13 | \( 1 - 55.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 32.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 129.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 249.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 388.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 109.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 309.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 137.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 648.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 715.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 655.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 308.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 694.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 957.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 460.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−8.623299981037998527518497046831, −7.78092924522972840501877045268, −6.71082357033612760827922089783, −6.02342770623761588161565205371, −5.32082970426158731366957767572, −4.48934719491285782467396663446, −3.40631721765144836077720989677, −1.78185322490518334567822606386, −1.33607931070403554767063231693, 0,
1.33607931070403554767063231693, 1.78185322490518334567822606386, 3.40631721765144836077720989677, 4.48934719491285782467396663446, 5.32082970426158731366957767572, 6.02342770623761588161565205371, 6.71082357033612760827922089783, 7.78092924522972840501877045268, 8.623299981037998527518497046831
