| L(s) = 1 | + 4.23·2-s + 9.94·4-s + 10.7·5-s + 10.6·7-s + 8.23·8-s + 45.5·10-s + 52.3·11-s − 79.0·13-s + 45.1·14-s − 44.6·16-s + 77.2·17-s + 50.7·19-s + 107.·20-s + 221.·22-s + 23·23-s − 9.13·25-s − 334.·26-s + 105.·28-s − 12.7·29-s − 12.4·31-s − 255.·32-s + 327.·34-s + 114.·35-s − 73.1·37-s + 215.·38-s + 88.6·40-s + 38.8·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.24·4-s + 0.962·5-s + 0.575·7-s + 0.363·8-s + 1.44·10-s + 1.43·11-s − 1.68·13-s + 0.861·14-s − 0.697·16-s + 1.10·17-s + 0.613·19-s + 1.19·20-s + 2.14·22-s + 0.208·23-s − 0.0731·25-s − 2.52·26-s + 0.714·28-s − 0.0816·29-s − 0.0719·31-s − 1.40·32-s + 1.65·34-s + 0.553·35-s − 0.324·37-s + 0.918·38-s + 0.350·40-s + 0.148·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.626798192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.626798192\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 4.23T + 8T^{2} \) |
| 5 | \( 1 - 10.7T + 125T^{2} \) |
| 7 | \( 1 - 10.6T + 343T^{2} \) |
| 11 | \( 1 - 52.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 50.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 12.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 73.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 38.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 614.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 269.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 534.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 838.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 448.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 628.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 925.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 963.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 133.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 778.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.60e3T + 9.12e5T^{2} \) |
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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
−12.08855167355999269154710049814, −11.49699498622575203397551005523, −9.978863069529243581178880349723, −9.219629653375036728487758313520, −7.52161288799196957430089060441, −6.40677883827513762853864538916, −5.42315269279550347711199280603, −4.58646275519706348734053763090, −3.18733937681174670635732516026, −1.76792535509813731041122588356,
1.76792535509813731041122588356, 3.18733937681174670635732516026, 4.58646275519706348734053763090, 5.42315269279550347711199280603, 6.40677883827513762853864538916, 7.52161288799196957430089060441, 9.219629653375036728487758313520, 9.978863069529243581178880349723, 11.49699498622575203397551005523, 12.08855167355999269154710049814
