| L(s) = 1 | − 2-s − 1.25·3-s + 4-s + 1.25·6-s + 7-s − 8-s − 1.42·9-s + 4.93·11-s − 1.25·12-s − 5.36·13-s − 14-s + 16-s − 17-s + 1.42·18-s − 6.80·19-s − 1.25·21-s − 4.93·22-s − 0.237·23-s + 1.25·24-s + 5.36·26-s + 5.55·27-s + 28-s − 6.61·29-s + 10.8·31-s − 32-s − 6.18·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.724·3-s + 0.5·4-s + 0.511·6-s + 0.377·7-s − 0.353·8-s − 0.475·9-s + 1.48·11-s − 0.362·12-s − 1.48·13-s − 0.267·14-s + 0.250·16-s − 0.242·17-s + 0.336·18-s − 1.56·19-s − 0.273·21-s − 1.05·22-s − 0.0495·23-s + 0.255·24-s + 1.05·26-s + 1.06·27-s + 0.188·28-s − 1.22·29-s + 1.94·31-s − 0.176·32-s − 1.07·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6919854613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6919854613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.25T + 3T^{2} \) |
| 11 | \( 1 - 4.93T + 11T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 + 0.237T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 2.85T + 37T^{2} \) |
| 41 | \( 1 + 8.16T + 41T^{2} \) |
| 43 | \( 1 - 1.25T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + 6.99T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 7.28T + 67T^{2} \) |
| 71 | \( 1 - 0.637T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 6.37T + 89T^{2} \) |
| 97 | \( 1 + 5.74T + 97T^{2} \) |
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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
−8.306315389842299976922874294328, −7.31105241341798147650026516053, −6.54828487848817614278271072537, −6.30095676113638985124258952898, −5.20397513329099063888090168454, −4.61747857710682477350051948805, −3.66788185043071243719557771959, −2.50506944741450033225994117867, −1.73126337399320236670837850776, −0.49802144660840286542274097438,
0.49802144660840286542274097438, 1.73126337399320236670837850776, 2.50506944741450033225994117867, 3.66788185043071243719557771959, 4.61747857710682477350051948805, 5.20397513329099063888090168454, 6.30095676113638985124258952898, 6.54828487848817614278271072537, 7.31105241341798147650026516053, 8.306315389842299976922874294328
