| L(s) = 1 | + 2.56·3-s − 5-s + 4.56·7-s + 3.56·9-s − 11-s − 1.12·13-s − 2.56·15-s − 7.68·17-s + 1.43·19-s + 11.6·21-s + 1.12·23-s + 25-s + 1.43·27-s + 8.56·29-s − 1.43·31-s − 2.56·33-s − 4.56·35-s + 7.43·37-s − 2.87·39-s − 12.2·41-s − 3.12·43-s − 3.56·45-s − 11.3·47-s + 13.8·49-s − 19.6·51-s + 9.68·53-s + 55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 0.447·5-s + 1.72·7-s + 1.18·9-s − 0.301·11-s − 0.311·13-s − 0.661·15-s − 1.86·17-s + 0.330·19-s + 2.54·21-s + 0.234·23-s + 0.200·25-s + 0.276·27-s + 1.58·29-s − 0.258·31-s − 0.445·33-s − 0.771·35-s + 1.22·37-s − 0.460·39-s − 1.91·41-s − 0.476·43-s − 0.530·45-s − 1.65·47-s + 1.97·49-s − 2.75·51-s + 1.33·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.284840721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.284840721\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 17 | \( 1 + 7.68T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 - 8.56T + 29T^{2} \) |
| 31 | \( 1 + 1.43T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 9.68T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 - 1.12T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 9.68T + 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
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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
−11.18993945231986005536130148749, −10.16934149193692953348718478314, −8.943436852860649154361555461279, −8.385717926981896984627248191318, −7.80066345135095209467174389858, −6.81167398105403617478743095018, −4.96565405982927898387085314234, −4.27501589463224150581335249703, −2.86225181205394860553829947599, −1.80645822875376192796414899877,
1.80645822875376192796414899877, 2.86225181205394860553829947599, 4.27501589463224150581335249703, 4.96565405982927898387085314234, 6.81167398105403617478743095018, 7.80066345135095209467174389858, 8.385717926981896984627248191318, 8.943436852860649154361555461279, 10.16934149193692953348718478314, 11.18993945231986005536130148749
