| L(s) = 1 | + 2·2-s + 8.12·3-s + 4·4-s + 16.6·5-s + 16.2·6-s − 7·7-s + 8·8-s + 38.9·9-s + 33.3·10-s + 70.2·11-s + 32.4·12-s − 14·14-s + 135.·15-s + 16·16-s + 32.8·17-s + 77.9·18-s + 47.6·19-s + 66.7·20-s − 56.8·21-s + 140.·22-s + 104.·23-s + 64.9·24-s + 153.·25-s + 97.2·27-s − 28·28-s − 217.·29-s + 271.·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.56·3-s + 0.5·4-s + 1.49·5-s + 1.10·6-s − 0.377·7-s + 0.353·8-s + 1.44·9-s + 1.05·10-s + 1.92·11-s + 0.781·12-s − 0.267·14-s + 2.33·15-s + 0.250·16-s + 0.468·17-s + 1.02·18-s + 0.574·19-s + 0.746·20-s − 0.590·21-s + 1.36·22-s + 0.947·23-s + 0.552·24-s + 1.23·25-s + 0.692·27-s − 0.188·28-s − 1.39·29-s + 1.65·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.24113247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.24113247\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 8.12T + 27T^{2} \) |
| 5 | \( 1 - 16.6T + 125T^{2} \) |
| 11 | \( 1 - 70.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 32.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 47.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 217.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 220.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 124.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 5.67T + 1.03e5T^{2} \) |
| 53 | \( 1 + 666.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 542.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 631.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 625.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 153.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 378.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 577.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 368.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 119.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 768.T + 9.12e5T^{2} \) |
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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
−9.038637232384831959321400985511, −7.76762435431468043865338958647, −7.02333635718653049768315408304, −6.27612952992114281118271480392, −5.57044076965848304687681308582, −4.45306542413310080572219729334, −3.46075032751710580089260165072, −3.04540749855757055461720311035, −1.81632158176703575920496528470, −1.46740150604001268380293177349,
1.46740150604001268380293177349, 1.81632158176703575920496528470, 3.04540749855757055461720311035, 3.46075032751710580089260165072, 4.45306542413310080572219729334, 5.57044076965848304687681308582, 6.27612952992114281118271480392, 7.02333635718653049768315408304, 7.76762435431468043865338958647, 9.038637232384831959321400985511
