| L(s) = 1 | − 2.79·2-s + 3-s + 5.79·4-s − 3·5-s − 2.79·6-s − 10.5·8-s + 9-s + 8.37·10-s − 11-s + 5.79·12-s − 13-s − 3·15-s + 17.9·16-s + 1.58·17-s − 2.79·18-s − 2.58·19-s − 17.3·20-s + 2.79·22-s + 3.58·23-s − 10.5·24-s + 4·25-s + 2.79·26-s + 27-s + 10.1·29-s + 8.37·30-s + 5.58·31-s − 28.9·32-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 0.577·3-s + 2.89·4-s − 1.34·5-s − 1.13·6-s − 3.74·8-s + 0.333·9-s + 2.64·10-s − 0.301·11-s + 1.67·12-s − 0.277·13-s − 0.774·15-s + 4.48·16-s + 0.383·17-s − 0.657·18-s − 0.592·19-s − 3.88·20-s + 0.595·22-s + 0.747·23-s − 2.16·24-s + 0.800·25-s + 0.547·26-s + 0.192·27-s + 1.88·29-s + 1.52·30-s + 1.00·31-s − 5.11·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 - 3.58T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 7.16T + 41T^{2} \) |
| 43 | \( 1 + 7.58T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 0.417T + 53T^{2} \) |
| 59 | \( 1 - 4.58T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 0.582T + 67T^{2} \) |
| 71 | \( 1 + 7.16T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 2.41T + 83T^{2} \) |
| 89 | \( 1 - 9.16T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−8.665221783134494462776258727655, −8.364012188149483828177457165658, −7.75635230286110231101839373515, −7.02059300056720139502231850646, −6.36020071806973725801500989607, −4.77064839903175161220750390788, −3.35580587489702250586874036971, −2.68822293149173797406090310108, −1.31423005067182296472482957375, 0,
1.31423005067182296472482957375, 2.68822293149173797406090310108, 3.35580587489702250586874036971, 4.77064839903175161220750390788, 6.36020071806973725801500989607, 7.02059300056720139502231850646, 7.75635230286110231101839373515, 8.364012188149483828177457165658, 8.665221783134494462776258727655
