| L(s) = 1 | − 1.33·3-s + 1.42·5-s − 1.22·9-s − 6.47·11-s − 13-s − 1.89·15-s − 5.48·17-s − 2.96·19-s + 4.94·23-s − 2.97·25-s + 5.63·27-s − 3.73·29-s − 3.98·31-s + 8.63·33-s − 6.58·37-s + 1.33·39-s − 10.3·41-s + 10.8·43-s − 1.73·45-s + 10.9·47-s + 7.32·51-s + 5.45·53-s − 9.20·55-s + 3.95·57-s + 2.14·59-s + 2.09·61-s − 1.42·65-s + ⋯ |
| L(s) = 1 | − 0.770·3-s + 0.635·5-s − 0.406·9-s − 1.95·11-s − 0.277·13-s − 0.489·15-s − 1.33·17-s − 0.679·19-s + 1.03·23-s − 0.595·25-s + 1.08·27-s − 0.692·29-s − 0.716·31-s + 1.50·33-s − 1.08·37-s + 0.213·39-s − 1.60·41-s + 1.65·43-s − 0.258·45-s + 1.60·47-s + 1.02·51-s + 0.748·53-s − 1.24·55-s + 0.523·57-s + 0.279·59-s + 0.268·61-s − 0.176·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6838649997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6838649997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 - 1.42T + 5T^{2} \) |
| 11 | \( 1 + 6.47T + 11T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 - 4.94T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 + 3.98T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 5.45T + 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 - 2.09T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 0.0739T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 5.97T + 89T^{2} \) |
| 97 | \( 1 + 1.49T + 97T^{2} \) |
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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
−8.303531063789854927603699172902, −7.36657513613208268690698667811, −6.75511020567216223507412991120, −5.89232249134720860985960649141, −5.30677028042127964297653853979, −4.92877957358795345396477999318, −3.76651540880206448924263821421, −2.54859075761296987543983583487, −2.13429291270869040821475698195, −0.43442021173416334801515558392,
0.43442021173416334801515558392, 2.13429291270869040821475698195, 2.54859075761296987543983583487, 3.76651540880206448924263821421, 4.92877957358795345396477999318, 5.30677028042127964297653853979, 5.89232249134720860985960649141, 6.75511020567216223507412991120, 7.36657513613208268690698667811, 8.303531063789854927603699172902
