| L(s) = 1 | + 0.414·2-s − 1.82·4-s + 5-s − 1.58·8-s + 0.414·10-s − 4.82·11-s − 0.828·13-s + 3·16-s − 0.828·17-s + 2.82·19-s − 1.82·20-s − 1.99·22-s + 2.41·23-s + 25-s − 0.343·26-s + 29-s + 6·31-s + 4.41·32-s − 0.343·34-s + 1.17·38-s − 1.58·40-s − 2.17·41-s + 6.41·43-s + 8.82·44-s + 0.999·46-s + 2·47-s + 0.414·50-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s + 0.447·5-s − 0.560·8-s + 0.130·10-s − 1.45·11-s − 0.229·13-s + 0.750·16-s − 0.200·17-s + 0.648·19-s − 0.408·20-s − 0.426·22-s + 0.503·23-s + 0.200·25-s − 0.0672·26-s + 0.185·29-s + 1.07·31-s + 0.780·32-s − 0.0588·34-s + 0.190·38-s − 0.250·40-s − 0.339·41-s + 0.978·43-s + 1.33·44-s + 0.147·46-s + 0.291·47-s + 0.0585·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.458220320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458220320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.41T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 6.41T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−9.081903751884427821723928215700, −8.295051618054720759783093170487, −7.64780582155133171512515723040, −6.63073814638768040818137817348, −5.59433109814177910829958867179, −5.14708635134894939708903978830, −4.34752971437381522600465814837, −3.21631757366628341768671418290, −2.39356275303105450313270662956, −0.75042265159337449102848382756,
0.75042265159337449102848382756, 2.39356275303105450313270662956, 3.21631757366628341768671418290, 4.34752971437381522600465814837, 5.14708635134894939708903978830, 5.59433109814177910829958867179, 6.63073814638768040818137817348, 7.64780582155133171512515723040, 8.295051618054720759783093170487, 9.081903751884427821723928215700
