| L(s) = 1 | − 2.69·3-s + 2.04·5-s + 7-s + 4.28·9-s + 0.288·13-s − 5.50·15-s − 6.39·17-s + 5.29·19-s − 2.69·21-s + 0.439·23-s − 0.838·25-s − 3.45·27-s − 5.56·29-s + 6.40·31-s + 2.04·35-s + 1.90·37-s − 0.778·39-s + 0.592·41-s + 7.16·43-s + 8.73·45-s − 12.0·47-s + 49-s + 17.2·51-s − 1.61·53-s − 14.2·57-s + 9.24·59-s − 9.87·61-s + ⋯ |
| L(s) = 1 | − 1.55·3-s + 0.912·5-s + 0.377·7-s + 1.42·9-s + 0.0800·13-s − 1.42·15-s − 1.55·17-s + 1.21·19-s − 0.588·21-s + 0.0915·23-s − 0.167·25-s − 0.664·27-s − 1.03·29-s + 1.15·31-s + 0.344·35-s + 0.313·37-s − 0.124·39-s + 0.0925·41-s + 1.09·43-s + 1.30·45-s − 1.75·47-s + 0.142·49-s + 2.41·51-s − 0.222·53-s − 1.89·57-s + 1.20·59-s − 1.26·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.280002423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280002423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 - 2.04T + 5T^{2} \) |
| 13 | \( 1 - 0.288T + 13T^{2} \) |
| 17 | \( 1 + 6.39T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 - 0.439T + 23T^{2} \) |
| 29 | \( 1 + 5.56T + 29T^{2} \) |
| 31 | \( 1 - 6.40T + 31T^{2} \) |
| 37 | \( 1 - 1.90T + 37T^{2} \) |
| 41 | \( 1 - 0.592T + 41T^{2} \) |
| 43 | \( 1 - 7.16T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 1.61T + 53T^{2} \) |
| 59 | \( 1 - 9.24T + 59T^{2} \) |
| 61 | \( 1 + 9.87T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 0.716T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 2.98T + 79T^{2} \) |
| 83 | \( 1 + 8.08T + 83T^{2} \) |
| 89 | \( 1 - 2.38T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−7.83222991635768254644730514852, −7.01665125526116894564528927129, −6.38093551471832657778036826891, −5.91437052860018538491012711580, −5.15741563673690424794122416437, −4.75510208698713149531434545984, −3.80011311980339637536883777140, −2.50748485616699691720089730449, −1.64134400921365750348637409826, −0.64687302981517371325833132807,
0.64687302981517371325833132807, 1.64134400921365750348637409826, 2.50748485616699691720089730449, 3.80011311980339637536883777140, 4.75510208698713149531434545984, 5.15741563673690424794122416437, 5.91437052860018538491012711580, 6.38093551471832657778036826891, 7.01665125526116894564528927129, 7.83222991635768254644730514852
