| L(s) = 1 | + 2·2-s + 2·4-s − 3·5-s + 7-s − 6·10-s + 6·11-s − 13-s + 2·14-s − 4·16-s + 17-s − 19-s − 6·20-s + 12·22-s + 5·23-s + 4·25-s − 2·26-s + 2·28-s + 29-s + 31-s − 8·32-s + 2·34-s − 3·35-s + 2·37-s − 2·38-s + 6·41-s − 5·43-s + 12·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.34·5-s + 0.377·7-s − 1.89·10-s + 1.80·11-s − 0.277·13-s + 0.534·14-s − 16-s + 0.242·17-s − 0.229·19-s − 1.34·20-s + 2.55·22-s + 1.04·23-s + 4/5·25-s − 0.392·26-s + 0.377·28-s + 0.185·29-s + 0.179·31-s − 1.41·32-s + 0.342·34-s − 0.507·35-s + 0.328·37-s − 0.324·38-s + 0.937·41-s − 0.762·43-s + 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.867174716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.867174716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−16.03225035995503, −15.25363201188224, −14.96546218411363, −14.51235362422660, −14.08574751430429, −13.37163462574870, −12.68110168601632, −12.21836283354497, −11.81049554367134, −11.24216506932701, −11.01373124218466, −9.802879639024815, −9.199231370023408, −8.554147537428590, −7.931839239241371, −7.107612360344052, −6.695459034728199, −6.045801313435439, −5.162124413096047, −4.537522267452194, −4.120678211162305, −3.500280387076914, −2.944863577618656, −1.787195673543911, −0.7091572639788803,
0.7091572639788803, 1.787195673543911, 2.944863577618656, 3.500280387076914, 4.120678211162305, 4.537522267452194, 5.162124413096047, 6.045801313435439, 6.695459034728199, 7.107612360344052, 7.931839239241371, 8.554147537428590, 9.199231370023408, 9.802879639024815, 11.01373124218466, 11.24216506932701, 11.81049554367134, 12.21836283354497, 12.68110168601632, 13.37163462574870, 14.08574751430429, 14.51235362422660, 14.96546218411363, 15.25363201188224, 16.03225035995503
