| L(s) = 1 | + 3.23·3-s + 2.81·5-s + 2.60·7-s + 7.44·9-s + 3.84·11-s − 0.602·13-s + 9.11·15-s − 5.45·17-s − 2.16·19-s + 8.41·21-s + 2.95·25-s + 14.3·27-s − 7.30·29-s − 4.94·31-s + 12.4·33-s + 7.34·35-s + 1.76·37-s − 1.94·39-s − 8.76·41-s + 2.93·43-s + 20.9·45-s + 6.55·47-s − 0.222·49-s − 17.6·51-s − 0.742·53-s + 10.8·55-s − 7.00·57-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 1.26·5-s + 0.983·7-s + 2.48·9-s + 1.15·11-s − 0.167·13-s + 2.35·15-s − 1.32·17-s − 0.497·19-s + 1.83·21-s + 0.590·25-s + 2.76·27-s − 1.35·29-s − 0.889·31-s + 2.16·33-s + 1.24·35-s + 0.290·37-s − 0.311·39-s − 1.36·41-s + 0.447·43-s + 3.12·45-s + 0.955·47-s − 0.0317·49-s − 2.46·51-s − 0.101·53-s + 1.46·55-s − 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.746016407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.746016407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 - 2.81T + 5T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 0.602T + 13T^{2} \) |
| 17 | \( 1 + 5.45T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 + 4.94T + 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 + 8.76T + 41T^{2} \) |
| 43 | \( 1 - 2.93T + 43T^{2} \) |
| 47 | \( 1 - 6.55T + 47T^{2} \) |
| 53 | \( 1 + 0.742T + 53T^{2} \) |
| 59 | \( 1 + 6.36T + 59T^{2} \) |
| 61 | \( 1 - 6.96T + 61T^{2} \) |
| 67 | \( 1 - 9.39T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 1.27T + 79T^{2} \) |
| 83 | \( 1 + 4.65T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 7.86T + 97T^{2} \) |
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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
−8.638003727029539075301957658095, −7.79460647668416019506862941768, −7.06933176596913886264467520320, −6.39865774830997766498773457239, −5.35027707922142673757686873613, −4.36501895518525830107322103843, −3.83305466768676275337211346775, −2.70130072566160335816077101558, −1.87145250682504495257250027929, −1.61969086792946089852460180414,
1.61969086792946089852460180414, 1.87145250682504495257250027929, 2.70130072566160335816077101558, 3.83305466768676275337211346775, 4.36501895518525830107322103843, 5.35027707922142673757686873613, 6.39865774830997766498773457239, 7.06933176596913886264467520320, 7.79460647668416019506862941768, 8.638003727029539075301957658095
