| L(s) = 1 | + 3.83·5-s + 3.19·7-s − 3.48·11-s + 5.48·13-s + 7.32·17-s − 3.48·19-s + 4.05·23-s + 9.68·25-s − 7.83·29-s + 6.97·31-s + 12.2·35-s − 37-s − 2.58·41-s − 4·43-s + 6.39·47-s + 3.21·49-s − 11.8·53-s − 13.3·55-s − 12.8·59-s + 6.58·61-s + 21.0·65-s + 3.37·67-s − 13.3·71-s + 10.4·73-s − 11.1·77-s + 5.27·79-s + 9.88·83-s + ⋯ |
| L(s) = 1 | + 1.71·5-s + 1.20·7-s − 1.05·11-s + 1.52·13-s + 1.77·17-s − 0.800·19-s + 0.844·23-s + 1.93·25-s − 1.45·29-s + 1.25·31-s + 2.07·35-s − 0.164·37-s − 0.403·41-s − 0.609·43-s + 0.932·47-s + 0.459·49-s − 1.63·53-s − 1.80·55-s − 1.66·59-s + 0.843·61-s + 2.60·65-s + 0.411·67-s − 1.58·71-s + 1.22·73-s − 1.27·77-s + 0.593·79-s + 1.08·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.637739155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.637739155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 3.83T + 5T^{2} \) |
| 7 | \( 1 - 3.19T + 7T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 - 5.48T + 13T^{2} \) |
| 17 | \( 1 - 7.32T + 17T^{2} \) |
| 19 | \( 1 + 3.48T + 19T^{2} \) |
| 23 | \( 1 - 4.05T + 23T^{2} \) |
| 29 | \( 1 + 7.83T + 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 41 | \( 1 + 2.58T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 6.58T + 61T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 5.27T + 79T^{2} \) |
| 83 | \( 1 - 9.88T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−8.137409115940839344415673387052, −7.68661463213789804147028746581, −6.52436652416317869702972342965, −5.94857127595246515907624199757, −5.29924308934925318312748339589, −4.86433031871118026622174598689, −3.59614220269243740113552461925, −2.68252171844312801670975787400, −1.73843282871180061210545089491, −1.17995356105643239685532551201,
1.17995356105643239685532551201, 1.73843282871180061210545089491, 2.68252171844312801670975787400, 3.59614220269243740113552461925, 4.86433031871118026622174598689, 5.29924308934925318312748339589, 5.94857127595246515907624199757, 6.52436652416317869702972342965, 7.68661463213789804147028746581, 8.137409115940839344415673387052
