| L(s) = 1 | − 1.09e5·2-s + 1.96e7·3-s + 3.34e9·4-s − 2.14e12·6-s − 7.24e12·7-s + 5.73e14·8-s − 5.17e15·9-s + 1.80e17·11-s + 6.55e16·12-s − 1.38e18·13-s + 7.91e17·14-s − 9.13e19·16-s + 6.46e19·17-s + 5.65e20·18-s + 1.46e21·19-s − 1.42e20·21-s − 1.97e22·22-s − 7.51e21·23-s + 1.12e22·24-s + 1.51e23·26-s − 2.10e23·27-s − 2.42e22·28-s − 3.42e23·29-s + 1.13e24·31-s + 5.05e24·32-s + 3.55e24·33-s − 7.06e24·34-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.263·3-s + 0.389·4-s − 0.310·6-s − 0.0823·7-s + 0.719·8-s − 0.930·9-s + 1.18·11-s + 0.102·12-s − 0.576·13-s + 0.0970·14-s − 1.23·16-s + 0.322·17-s + 1.09·18-s + 1.16·19-s − 0.0216·21-s − 1.39·22-s − 0.255·23-s + 0.189·24-s + 0.679·26-s − 0.508·27-s − 0.0320·28-s − 0.254·29-s + 0.281·31-s + 0.738·32-s + 0.312·33-s − 0.379·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(1.032858882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.032858882\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 1.09e5T + 8.58e9T^{2} \) |
| 3 | \( 1 - 1.96e7T + 5.55e15T^{2} \) |
| 7 | \( 1 + 7.24e12T + 7.73e27T^{2} \) |
| 11 | \( 1 - 1.80e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 1.38e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 6.46e19T + 4.02e40T^{2} \) |
| 19 | \( 1 - 1.46e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 7.51e21T + 8.65e44T^{2} \) |
| 29 | \( 1 + 3.42e23T + 1.81e48T^{2} \) |
| 31 | \( 1 - 1.13e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 2.61e25T + 5.63e51T^{2} \) |
| 41 | \( 1 - 1.74e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.37e27T + 8.02e53T^{2} \) |
| 47 | \( 1 - 3.57e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 3.47e27T + 7.96e56T^{2} \) |
| 59 | \( 1 + 1.06e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 3.27e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 5.06e29T + 1.82e60T^{2} \) |
| 71 | \( 1 + 2.35e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 2.40e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 2.85e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 1.04e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 2.05e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 7.08e32T + 3.65e65T^{2} \) |
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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
−11.08741206972895221661430950108, −9.734582458330609627601216879164, −9.098171362451881311507623931437, −8.053881086087529819315684017333, −7.04446113585258274647110419483, −5.59999627825043240081021522163, −4.13292660601073185779629111779, −2.80255475973463984986689312784, −1.51582818424648519883358913219, −0.54963543261609962836632271496,
0.54963543261609962836632271496, 1.51582818424648519883358913219, 2.80255475973463984986689312784, 4.13292660601073185779629111779, 5.59999627825043240081021522163, 7.04446113585258274647110419483, 8.053881086087529819315684017333, 9.098171362451881311507623931437, 9.734582458330609627601216879164, 11.08741206972895221661430950108
