| L(s) = 1 | − 2.32·3-s + 5-s − 4.78·7-s + 2.40·9-s + 0.601·13-s − 2.32·15-s + 2.64·17-s − 2.60·19-s + 11.1·21-s + 5.17·23-s + 25-s + 1.38·27-s − 5.44·29-s − 10.4·31-s − 4.78·35-s + 9.39·37-s − 1.39·39-s + 4.54·41-s + 3.48·43-s + 2.40·45-s + 5.07·47-s + 15.9·49-s − 6.15·51-s + 3.76·53-s + 6.04·57-s + 11.7·59-s − 12.6·61-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.447·5-s − 1.80·7-s + 0.801·9-s + 0.166·13-s − 0.600·15-s + 0.642·17-s − 0.596·19-s + 2.42·21-s + 1.07·23-s + 0.200·25-s + 0.266·27-s − 1.01·29-s − 1.87·31-s − 0.809·35-s + 1.54·37-s − 0.223·39-s + 0.710·41-s + 0.531·43-s + 0.358·45-s + 0.740·47-s + 2.27·49-s − 0.862·51-s + 0.517·53-s + 0.801·57-s + 1.53·59-s − 1.61·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.32T + 3T^{2} \) |
| 7 | \( 1 + 4.78T + 7T^{2} \) |
| 13 | \( 1 - 0.601T + 13T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 - 5.17T + 23T^{2} \) |
| 29 | \( 1 + 5.44T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 - 4.54T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 - 5.07T + 47T^{2} \) |
| 53 | \( 1 - 3.76T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 - 0.346T + 71T^{2} \) |
| 73 | \( 1 + 3.68T + 73T^{2} \) |
| 79 | \( 1 - 8.05T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 0.462T + 89T^{2} \) |
| 97 | \( 1 - 7.97T + 97T^{2} \) |
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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
−7.58454133826566059421951431273, −7.00991038972219723378947892975, −6.23006547853297793292165745494, −5.87256117532739897690378118496, −5.28484396848594803496648345588, −4.19144598231374951704796550641, −3.37222660974894868823018848867, −2.45450545892824961281589036281, −1.01155350655029705685685737547, 0,
1.01155350655029705685685737547, 2.45450545892824961281589036281, 3.37222660974894868823018848867, 4.19144598231374951704796550641, 5.28484396848594803496648345588, 5.87256117532739897690378118496, 6.23006547853297793292165745494, 7.00991038972219723378947892975, 7.58454133826566059421951431273
