| L(s) = 1 | + 1.75·5-s − 6.54·11-s − 3.60i·13-s − 1.92·25-s + 10.1i·41-s + 12.4i·43-s + 9.33i·47-s + 7·49-s − 11.4·55-s + 0.469·59-s + 7.21i·61-s − 6.31i·65-s + 4.92i·71-s − 10.3·79-s − 12.6·83-s + ⋯ |
| L(s) = 1 | + 0.783·5-s − 1.97·11-s − 0.999i·13-s − 0.385·25-s + 1.58i·41-s + 1.90i·43-s + 1.36i·47-s + 49-s − 1.54·55-s + 0.0611·59-s + 0.923i·61-s − 0.783i·65-s + 0.584i·71-s − 1.16·79-s − 1.38·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7719485538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7719485538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + 3.60iT \) |
| good | 5 | \( 1 - 1.75T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 6.54T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 10.1iT - 41T^{2} \) |
| 43 | \( 1 - 12.4iT - 43T^{2} \) |
| 47 | \( 1 - 9.33iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 0.469T + 59T^{2} \) |
| 61 | \( 1 - 7.21iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 4.92iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 18.3iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.652913812726372926977752906597, −7.916250012684861594525901886260, −7.54144435370663479010637599362, −6.37815123841420237665788376724, −5.68633912807337922845782471438, −5.19487019784420696378486896982, −4.30279337189573326429601016319, −2.90436620163659937029450922054, −2.60800076991114118504629517846, −1.26877853211317785333352721862,
0.21328213861212799566205555062, 1.92032784885752289743081558132, 2.41138262642296149457056765752, 3.54597618290620838440009064200, 4.54875974277053161993127855682, 5.46116098291667833167137660235, 5.76233382005202129360027050114, 6.93129940300376375558826955470, 7.40122079143956895039765094620, 8.372916127857210711545818137109
