| L(s) = 1 | − 2.33·3-s − 0.450·5-s + 7-s + 2.42·9-s + 11-s + 13-s + 1.05·15-s − 4.58·17-s − 1.00·19-s − 2.33·21-s − 2.87·23-s − 4.79·25-s + 1.32·27-s + 5.30·29-s − 0.0259·31-s − 2.33·33-s − 0.450·35-s + 6.00·37-s − 2.33·39-s − 0.530·41-s − 6.30·43-s − 1.09·45-s + 1.34·47-s + 49-s + 10.6·51-s − 5.06·53-s − 0.450·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s − 0.201·5-s + 0.377·7-s + 0.809·9-s + 0.301·11-s + 0.277·13-s + 0.271·15-s − 1.11·17-s − 0.229·19-s − 0.508·21-s − 0.599·23-s − 0.959·25-s + 0.255·27-s + 0.984·29-s − 0.00466·31-s − 0.405·33-s − 0.0761·35-s + 0.987·37-s − 0.373·39-s − 0.0828·41-s − 0.961·43-s − 0.163·45-s + 0.196·47-s + 0.142·49-s + 1.49·51-s − 0.695·53-s − 0.0607·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8408629465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8408629465\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.33T + 3T^{2} \) |
| 5 | \( 1 + 0.450T + 5T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 + 1.00T + 19T^{2} \) |
| 23 | \( 1 + 2.87T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 + 0.0259T + 31T^{2} \) |
| 37 | \( 1 - 6.00T + 37T^{2} \) |
| 41 | \( 1 + 0.530T + 41T^{2} \) |
| 43 | \( 1 + 6.30T + 43T^{2} \) |
| 47 | \( 1 - 1.34T + 47T^{2} \) |
| 53 | \( 1 + 5.06T + 53T^{2} \) |
| 59 | \( 1 - 0.893T + 59T^{2} \) |
| 61 | \( 1 - 9.35T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 4.88T + 71T^{2} \) |
| 73 | \( 1 - 9.71T + 73T^{2} \) |
| 79 | \( 1 + 7.26T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 2.17T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−7.85438059383669434275901554462, −6.85933592292992397753499956828, −6.41676682590936717834123344382, −5.83269176293535715856140096993, −5.04168924477030781383710559449, −4.44163940722464815022935546978, −3.78620137538086576339349232348, −2.55938906126441837948039605562, −1.57485287582550099790563869220, −0.49815946395712688486196444512,
0.49815946395712688486196444512, 1.57485287582550099790563869220, 2.55938906126441837948039605562, 3.78620137538086576339349232348, 4.44163940722464815022935546978, 5.04168924477030781383710559449, 5.83269176293535715856140096993, 6.41676682590936717834123344382, 6.85933592292992397753499956828, 7.85438059383669434275901554462
