L(s) = 1 | + 4.88·2-s + 3·3-s + 15.9·4-s + 14.6·6-s − 7·7-s + 38.6·8-s + 9·9-s + 54.9·11-s + 47.7·12-s + 49.7·13-s − 34.2·14-s + 61.7·16-s − 133.·17-s + 44.0·18-s + 138.·19-s − 21·21-s + 268.·22-s − 7.32·23-s + 115.·24-s + 243.·26-s + 27·27-s − 111.·28-s + 87.2·29-s − 209.·31-s − 7.32·32-s + 164.·33-s − 653.·34-s + ⋯ |
L(s) = 1 | + 1.72·2-s + 0.577·3-s + 1.98·4-s + 0.998·6-s − 0.377·7-s + 1.70·8-s + 0.333·9-s + 1.50·11-s + 1.14·12-s + 1.06·13-s − 0.653·14-s + 0.964·16-s − 1.90·17-s + 0.576·18-s + 1.67·19-s − 0.218·21-s + 2.60·22-s − 0.0664·23-s + 0.986·24-s + 1.83·26-s + 0.192·27-s − 0.751·28-s + 0.558·29-s − 1.21·31-s − 0.0404·32-s + 0.868·33-s − 3.29·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(7.208358841\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.208358841\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
good | 2 | \( 1 - 4.88T + 8T^{2} \) |
| 11 | \( 1 - 54.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 49.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 133.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 138.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.32T + 1.21e4T^{2} \) |
| 29 | \( 1 - 87.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 67.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 77.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 4.97T + 1.03e5T^{2} \) |
| 53 | \( 1 - 53.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 683.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 26.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 149.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 6.15T + 3.57e5T^{2} \) |
| 73 | \( 1 + 294.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 938.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 784.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 275.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−10.92290017569781486699172988488, −9.437142264859819901687027020668, −8.783942544354609298711670297400, −7.28573962829621897031868380717, −6.56282710405041460960802856145, −5.78114807145615486758374856344, −4.44802316259407909141918531504, −3.79449963010170565958273280890, −2.88187889879734861122739696310, −1.53067848342335046443945344614,
1.53067848342335046443945344614, 2.88187889879734861122739696310, 3.79449963010170565958273280890, 4.44802316259407909141918531504, 5.78114807145615486758374856344, 6.56282710405041460960802856145, 7.28573962829621897031868380717, 8.783942544354609298711670297400, 9.437142264859819901687027020668, 10.92290017569781486699172988488