| L(s) = 1 | + 2·2-s + 8.77·3-s + 4·4-s − 17.3·5-s + 17.5·6-s − 26.0·7-s + 8·8-s + 49.9·9-s − 34.6·10-s − 4.22·11-s + 35.0·12-s + 64.0·13-s − 52.1·14-s − 151.·15-s + 16·16-s − 48.5·17-s + 99.8·18-s + 19·19-s − 69.2·20-s − 228.·21-s − 8.45·22-s + 92.0·23-s + 70.1·24-s + 174.·25-s + 128.·26-s + 201.·27-s − 104.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.68·3-s + 0.5·4-s − 1.54·5-s + 1.19·6-s − 1.40·7-s + 0.353·8-s + 1.84·9-s − 1.09·10-s − 0.115·11-s + 0.844·12-s + 1.36·13-s − 0.996·14-s − 2.61·15-s + 0.250·16-s − 0.692·17-s + 1.30·18-s + 0.229·19-s − 0.774·20-s − 2.37·21-s − 0.0819·22-s + 0.834·23-s + 0.596·24-s + 1.39·25-s + 0.966·26-s + 1.43·27-s − 0.704·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.226196849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.226196849\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 8.77T + 27T^{2} \) |
| 5 | \( 1 + 17.3T + 125T^{2} \) |
| 7 | \( 1 + 26.0T + 343T^{2} \) |
| 11 | \( 1 + 4.22T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 92.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 88.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 81.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 23.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 497.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 36.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 630.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 595.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 597.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 427.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 493.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 921.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−15.72995924244702178401287918599, −14.71090850537627076555606662756, −13.38775329037008038079963478557, −12.72576187683773906907546760818, −11.08453363412813674368454222799, −9.261357644577212813933289374130, −8.084959882271028323143027320107, −6.83945744702810489928646858932, −3.93149329831377357804181671820, −3.16303426781052515390090952770,
3.16303426781052515390090952770, 3.93149329831377357804181671820, 6.83945744702810489928646858932, 8.084959882271028323143027320107, 9.261357644577212813933289374130, 11.08453363412813674368454222799, 12.72576187683773906907546760818, 13.38775329037008038079963478557, 14.71090850537627076555606662756, 15.72995924244702178401287918599
