| L(s) = 1 | − 1.73·2-s + 2.52·3-s + 0.999·4-s − 4.37·6-s + 1.58·7-s + 1.73·8-s + 3.37·9-s + 6.37·11-s + 2.52·12-s − 0.939·13-s − 2.74·14-s − 5·16-s − 5.04·17-s − 5.84·18-s + 4·19-s + 4·21-s − 11.0·22-s − 3.46·23-s + 4.37·24-s + 1.62·26-s + 0.939·27-s + 1.58·28-s − 29-s + 2.37·31-s + 5.19·32-s + 16.0·33-s + 8.74·34-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 1.45·3-s + 0.499·4-s − 1.78·6-s + 0.598·7-s + 0.612·8-s + 1.12·9-s + 1.92·11-s + 0.728·12-s − 0.260·13-s − 0.733·14-s − 1.25·16-s − 1.22·17-s − 1.37·18-s + 0.917·19-s + 0.872·21-s − 2.35·22-s − 0.722·23-s + 0.892·24-s + 0.319·26-s + 0.180·27-s + 0.299·28-s − 0.185·29-s + 0.426·31-s + 0.918·32-s + 2.80·33-s + 1.49·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.477703066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.477703066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 - 2.52T + 3T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 + 0.939T + 13T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 6.74T + 41T^{2} \) |
| 43 | \( 1 + 5.69T + 43T^{2} \) |
| 47 | \( 1 + 5.69T + 47T^{2} \) |
| 53 | \( 1 + 0.939T + 53T^{2} \) |
| 59 | \( 1 - 0.744T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 8.51T + 67T^{2} \) |
| 71 | \( 1 + 4.74T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 5.62T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−9.731366739806023087332957620799, −9.511362571701865824979391997363, −8.699242421914448095050877816279, −8.143527870268631427992448053775, −7.35145469917342474814226061172, −6.43392109823624342081227793492, −4.59959358451985462742552877344, −3.78773641512783063365353027357, −2.31329215939164738681910428802, −1.32139496593749583994855245692,
1.32139496593749583994855245692, 2.31329215939164738681910428802, 3.78773641512783063365353027357, 4.59959358451985462742552877344, 6.43392109823624342081227793492, 7.35145469917342474814226061172, 8.143527870268631427992448053775, 8.699242421914448095050877816279, 9.511362571701865824979391997363, 9.731366739806023087332957620799
