| L(s) = 1 | − 2.92·3-s + 3.33·5-s + 5.58·9-s + 2.02·11-s + 3.10·13-s − 9.76·15-s + 7.91·17-s − 2.17·19-s − 4.75·23-s + 6.10·25-s − 7.56·27-s + 3.85·29-s + 10.6·31-s − 5.94·33-s + 6.58·37-s − 9.10·39-s + 41-s + 4.80·43-s + 18.6·45-s − 4.03·47-s − 23.1·51-s − 1.94·53-s + 6.76·55-s + 6.38·57-s − 4.75·59-s − 2.89·61-s + 10.3·65-s + ⋯ |
| L(s) = 1 | − 1.69·3-s + 1.49·5-s + 1.86·9-s + 0.611·11-s + 0.862·13-s − 2.52·15-s + 1.91·17-s − 0.500·19-s − 0.990·23-s + 1.22·25-s − 1.45·27-s + 0.716·29-s + 1.91·31-s − 1.03·33-s + 1.08·37-s − 1.45·39-s + 0.156·41-s + 0.733·43-s + 2.77·45-s − 0.589·47-s − 3.24·51-s − 0.266·53-s + 0.911·55-s + 0.845·57-s − 0.618·59-s − 0.370·61-s + 1.28·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.034071453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.034071453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 2.92T + 3T^{2} \) |
| 5 | \( 1 - 3.33T + 5T^{2} \) |
| 11 | \( 1 - 2.02T + 11T^{2} \) |
| 13 | \( 1 - 3.10T + 13T^{2} \) |
| 17 | \( 1 - 7.91T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 + 4.75T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 6.58T + 37T^{2} \) |
| 43 | \( 1 - 4.80T + 43T^{2} \) |
| 47 | \( 1 + 4.03T + 47T^{2} \) |
| 53 | \( 1 + 1.94T + 53T^{2} \) |
| 59 | \( 1 + 4.75T + 59T^{2} \) |
| 61 | \( 1 + 2.89T + 61T^{2} \) |
| 67 | \( 1 - 5.97T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 9.24T + 73T^{2} \) |
| 79 | \( 1 + 0.320T + 79T^{2} \) |
| 83 | \( 1 - 3.69T + 83T^{2} \) |
| 89 | \( 1 + 8.96T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−7.76320569075034775450003867708, −6.66367853020382633611920162572, −6.17821302046947611950376449444, −5.99162556990168709222888793217, −5.24270651171888901579667815582, −4.57334345666562291311843201601, −3.66923624399222566219068459732, −2.48510818608992314013434872527, −1.36080858171967737786895083088, −0.911941813392380263609007453269,
0.911941813392380263609007453269, 1.36080858171967737786895083088, 2.48510818608992314013434872527, 3.66923624399222566219068459732, 4.57334345666562291311843201601, 5.24270651171888901579667815582, 5.99162556990168709222888793217, 6.17821302046947611950376449444, 6.66367853020382633611920162572, 7.76320569075034775450003867708
