| L(s) = 1 | − 2.19·3-s + 0.635·5-s + 1.83·9-s + 11-s + 1.80·13-s − 1.39·15-s + 2.83·17-s − 5.56·19-s − 2.16·23-s − 4.59·25-s + 2.56·27-s − 10.4·29-s + 6.43·31-s − 2.19·33-s + 6.06·37-s − 3.96·39-s − 7.53·41-s + 4.86·43-s + 1.16·45-s − 2.83·47-s − 6.23·51-s + 7.46·53-s + 0.635·55-s + 12.2·57-s + 11.8·59-s + 4.33·61-s + 1.14·65-s + ⋯ |
| L(s) = 1 | − 1.26·3-s + 0.284·5-s + 0.611·9-s + 0.301·11-s + 0.499·13-s − 0.360·15-s + 0.687·17-s − 1.27·19-s − 0.451·23-s − 0.919·25-s + 0.493·27-s − 1.93·29-s + 1.15·31-s − 0.382·33-s + 0.997·37-s − 0.634·39-s − 1.17·41-s + 0.742·43-s + 0.173·45-s − 0.413·47-s − 0.872·51-s + 1.02·53-s + 0.0856·55-s + 1.62·57-s + 1.53·59-s + 0.554·61-s + 0.141·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.034558231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.034558231\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.19T + 3T^{2} \) |
| 5 | \( 1 - 0.635T + 5T^{2} \) |
| 13 | \( 1 - 1.80T + 13T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 - 6.06T + 37T^{2} \) |
| 41 | \( 1 + 7.53T + 41T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 + 2.83T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 4.33T + 61T^{2} \) |
| 67 | \( 1 - 1.60T + 67T^{2} \) |
| 71 | \( 1 + 4.29T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 4.76T + 79T^{2} \) |
| 83 | \( 1 + 9.23T + 83T^{2} \) |
| 89 | \( 1 + 0.364T + 89T^{2} \) |
| 97 | \( 1 - 2.59T + 97T^{2} \) |
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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
−7.75880824666436903907500815235, −6.79034258083939378738798129927, −6.33646532118601794368288586044, −5.67921905424836474925773554258, −5.28715091171058682090051519233, −4.23375912826409962616916066837, −3.75968886755393646503923257028, −2.50448415509251224262496978096, −1.59086349028464456926511351702, −0.54128583886214059430583331692,
0.54128583886214059430583331692, 1.59086349028464456926511351702, 2.50448415509251224262496978096, 3.75968886755393646503923257028, 4.23375912826409962616916066837, 5.28715091171058682090051519233, 5.67921905424836474925773554258, 6.33646532118601794368288586044, 6.79034258083939378738798129927, 7.75880824666436903907500815235
