| L(s) = 1 | − 0.306·2-s + 1.50·3-s − 1.90·4-s − 0.333·5-s − 0.461·6-s + 1.19·8-s − 0.734·9-s + 0.102·10-s + 5.36·11-s − 2.86·12-s − 6.79·13-s − 0.502·15-s + 3.44·16-s + 1.40·17-s + 0.225·18-s + 2.64·19-s + 0.636·20-s − 1.64·22-s + 2.49·23-s + 1.80·24-s − 4.88·25-s + 2.08·26-s − 5.62·27-s + 8.50·29-s + 0.154·30-s − 1.13·31-s − 3.45·32-s + ⋯ |
| L(s) = 1 | − 0.216·2-s + 0.868·3-s − 0.952·4-s − 0.149·5-s − 0.188·6-s + 0.423·8-s − 0.244·9-s + 0.0323·10-s + 1.61·11-s − 0.828·12-s − 1.88·13-s − 0.129·15-s + 0.861·16-s + 0.339·17-s + 0.0531·18-s + 0.606·19-s + 0.142·20-s − 0.351·22-s + 0.520·23-s + 0.368·24-s − 0.977·25-s + 0.408·26-s − 1.08·27-s + 1.57·29-s + 0.0281·30-s − 0.203·31-s − 0.610·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.576570351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576570351\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.306T + 2T^{2} \) |
| 3 | \( 1 - 1.50T + 3T^{2} \) |
| 5 | \( 1 + 0.333T + 5T^{2} \) |
| 11 | \( 1 - 5.36T + 11T^{2} \) |
| 13 | \( 1 + 6.79T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 - 8.50T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 43 | \( 1 + 4.95T + 43T^{2} \) |
| 47 | \( 1 - 8.99T + 47T^{2} \) |
| 53 | \( 1 - 6.86T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 - 3.32T + 67T^{2} \) |
| 71 | \( 1 + 0.907T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 4.90T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 7.42T + 97T^{2} \) |
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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
−9.173756796262757366146768520167, −8.511565208754698141743262936113, −7.73466105588224363783897611687, −7.10606200157580208685765942468, −5.92982591488984135009918864912, −4.94095534321940118352464300149, −4.16659672080587017503334123165, −3.33893712946738174954054095335, −2.30723762245644477213208835110, −0.847947335412117884638214429095,
0.847947335412117884638214429095, 2.30723762245644477213208835110, 3.33893712946738174954054095335, 4.16659672080587017503334123165, 4.94095534321940118352464300149, 5.92982591488984135009918864912, 7.10606200157580208685765942468, 7.73466105588224363783897611687, 8.511565208754698141743262936113, 9.173756796262757366146768520167
