| L(s) = 1 | + 2.90·3-s + 5-s − 0.903·7-s + 5.42·9-s + 5.52·11-s − 0.622·13-s + 2.90·15-s + 3.52·17-s + 1.09·19-s − 2.62·21-s − 5.33·23-s + 25-s + 7.05·27-s − 29-s + 1.65·31-s + 16.0·33-s − 0.903·35-s − 2.28·37-s − 1.80·39-s − 7.67·41-s + 1.09·43-s + 5.42·45-s + 1.65·47-s − 6.18·49-s + 10.2·51-s − 2.42·53-s + 5.52·55-s + ⋯ |
| L(s) = 1 | + 1.67·3-s + 0.447·5-s − 0.341·7-s + 1.80·9-s + 1.66·11-s − 0.172·13-s + 0.749·15-s + 0.855·17-s + 0.251·19-s − 0.572·21-s − 1.11·23-s + 0.200·25-s + 1.35·27-s − 0.185·29-s + 0.297·31-s + 2.79·33-s − 0.152·35-s − 0.374·37-s − 0.289·39-s − 1.19·41-s + 0.167·43-s + 0.809·45-s + 0.241·47-s − 0.883·49-s + 1.43·51-s − 0.333·53-s + 0.745·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.862830387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.862830387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 + 0.903T + 7T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 13 | \( 1 + 0.622T + 13T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 - 1.09T + 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 + 2.28T + 37T^{2} \) |
| 41 | \( 1 + 7.67T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 2.42T + 53T^{2} \) |
| 59 | \( 1 + 9.28T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 7.18T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 7.95T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−9.047067250519941058272575151335, −8.315105351374647657551037230892, −7.64550900838644964991957735691, −6.76187771708589136853220380264, −6.09188227127690924574333669058, −4.82022133134598833559201925454, −3.73667721452810618475048782909, −3.35211118480778182054904795429, −2.19452430266067776964663909734, −1.36879956828588715663350878648,
1.36879956828588715663350878648, 2.19452430266067776964663909734, 3.35211118480778182054904795429, 3.73667721452810618475048782909, 4.82022133134598833559201925454, 6.09188227127690924574333669058, 6.76187771708589136853220380264, 7.64550900838644964991957735691, 8.315105351374647657551037230892, 9.047067250519941058272575151335
