| L(s) = 1 | + 5.86·3-s + 5·5-s + 34.7·7-s + 7.34·9-s + 70.0·11-s − 10.1·13-s + 29.3·15-s + 104.·17-s + 162.·19-s + 203.·21-s − 161.·23-s + 25·25-s − 115.·27-s − 29·29-s − 315.·31-s + 410.·33-s + 173.·35-s − 57.1·37-s − 59.7·39-s + 180.·41-s − 304.·43-s + 36.7·45-s − 268.·47-s + 867.·49-s + 615.·51-s − 552.·53-s + 350.·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s + 0.447·5-s + 1.87·7-s + 0.271·9-s + 1.92·11-s − 0.217·13-s + 0.504·15-s + 1.49·17-s + 1.95·19-s + 2.11·21-s − 1.46·23-s + 0.200·25-s − 0.821·27-s − 0.185·29-s − 1.82·31-s + 2.16·33-s + 0.840·35-s − 0.253·37-s − 0.245·39-s + 0.685·41-s − 1.07·43-s + 0.121·45-s − 0.832·47-s + 2.52·49-s + 1.68·51-s − 1.43·53-s + 0.859·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.200069442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.200069442\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 5.86T + 27T^{2} \) |
| 7 | \( 1 - 34.7T + 343T^{2} \) |
| 11 | \( 1 - 70.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 162.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 315.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 57.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 180.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 304.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 268.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 552.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 330.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 266.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 161.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 49.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 612.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 701.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.20e3T + 9.12e5T^{2} \) |
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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
−9.393299696417880895162036432887, −8.578428815313720289753652413467, −7.81296962118849967302686481955, −7.33129442192662607429070329738, −5.89651058284467259215115410069, −5.14850697573263067702296159322, −3.99007393524597543987701420604, −3.22541424449192871775818944751, −1.74976667164370701495059073951, −1.40030347431421034848539053090,
1.40030347431421034848539053090, 1.74976667164370701495059073951, 3.22541424449192871775818944751, 3.99007393524597543987701420604, 5.14850697573263067702296159322, 5.89651058284467259215115410069, 7.33129442192662607429070329738, 7.81296962118849967302686481955, 8.578428815313720289753652413467, 9.393299696417880895162036432887
