| L(s) = 1 | + 2-s + 2.29·3-s + 4-s + 2.29·6-s − 1.25·7-s + 8-s + 2.25·9-s + 2·11-s + 2.29·12-s + 13-s − 1.25·14-s + 16-s + 4.80·17-s + 2.25·18-s − 5.09·19-s − 2.87·21-s + 2·22-s − 2.58·23-s + 2.29·24-s + 26-s − 1.70·27-s − 1.25·28-s + 5.09·29-s + 8.58·31-s + 32-s + 4.58·33-s + 4.80·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.935·6-s − 0.474·7-s + 0.353·8-s + 0.751·9-s + 0.603·11-s + 0.661·12-s + 0.277·13-s − 0.335·14-s + 0.250·16-s + 1.16·17-s + 0.531·18-s − 1.16·19-s − 0.627·21-s + 0.426·22-s − 0.538·23-s + 0.467·24-s + 0.196·26-s − 0.328·27-s − 0.237·28-s + 0.946·29-s + 1.54·31-s + 0.176·32-s + 0.798·33-s + 0.823·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.342916923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.342916923\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.29T + 3T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 + 5.09T + 19T^{2} \) |
| 23 | \( 1 + 2.58T + 23T^{2} \) |
| 29 | \( 1 - 5.09T + 29T^{2} \) |
| 31 | \( 1 - 8.58T + 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 41 | \( 1 + 9.67T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 - 5.38T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 - 6.26T + 97T^{2} \) |
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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
−10.31070904435950212812600412734, −9.780154197229813220997409135135, −8.551098722056452767692876897214, −8.170853712915319883613391038322, −6.89503630605462440704286405668, −6.18108686749386443483918385014, −4.82216061412183189967938018675, −3.67191952564387840603540066156, −3.07107705661602441580678764547, −1.79092080666297334905946892235,
1.79092080666297334905946892235, 3.07107705661602441580678764547, 3.67191952564387840603540066156, 4.82216061412183189967938018675, 6.18108686749386443483918385014, 6.89503630605462440704286405668, 8.170853712915319883613391038322, 8.551098722056452767692876897214, 9.780154197229813220997409135135, 10.31070904435950212812600412734
