| L(s) = 1 | − 1.80·2-s + 1.24·4-s − 1.44·5-s + 3.44·7-s + 1.35·8-s + 2.60·10-s + 5.18·11-s − 6.20·14-s − 4.93·16-s + 0.753·17-s + 7.96·19-s − 1.80·20-s − 9.34·22-s + 2.82·23-s − 2.91·25-s + 4.29·28-s + 3.91·29-s − 4.89·31-s + 6.18·32-s − 1.35·34-s − 4.97·35-s + 6.24·37-s − 14.3·38-s − 1.96·40-s − 1.80·41-s − 7.09·43-s + 6.46·44-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.623·4-s − 0.646·5-s + 1.30·7-s + 0.479·8-s + 0.823·10-s + 1.56·11-s − 1.65·14-s − 1.23·16-s + 0.182·17-s + 1.82·19-s − 0.402·20-s − 1.99·22-s + 0.589·23-s − 0.582·25-s + 0.811·28-s + 0.726·29-s − 0.880·31-s + 1.09·32-s − 0.232·34-s − 0.841·35-s + 1.02·37-s − 2.32·38-s − 0.310·40-s − 0.281·41-s − 1.08·43-s + 0.974·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.020829795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.020829795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 - 5.18T + 11T^{2} \) |
| 17 | \( 1 - 0.753T + 17T^{2} \) |
| 19 | \( 1 - 7.96T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 + 1.87T + 59T^{2} \) |
| 61 | \( 1 - 3.34T + 61T^{2} \) |
| 67 | \( 1 + 4.54T + 67T^{2} \) |
| 71 | \( 1 - 9.11T + 71T^{2} \) |
| 73 | \( 1 - 2.95T + 73T^{2} \) |
| 79 | \( 1 + 9.43T + 79T^{2} \) |
| 83 | \( 1 - 6.46T + 83T^{2} \) |
| 89 | \( 1 + 1.15T + 89T^{2} \) |
| 97 | \( 1 - 8.65T + 97T^{2} \) |
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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
−9.380276944758473757453798599194, −8.663699059686482958945018162911, −7.950360177606257472205779809633, −7.43867106535463434836179979318, −6.59718434969426991753084786331, −5.22976804642517915656391175308, −4.43239329480096584462974720110, −3.43042286262213563840377015039, −1.73175090359848183878626879465, −0.967932248701381155451065392591,
0.967932248701381155451065392591, 1.73175090359848183878626879465, 3.43042286262213563840377015039, 4.43239329480096584462974720110, 5.22976804642517915656391175308, 6.59718434969426991753084786331, 7.43867106535463434836179979318, 7.950360177606257472205779809633, 8.663699059686482958945018162911, 9.380276944758473757453798599194
