| L(s) = 1 | − 6.26e4·2-s − 3.24e7·3-s − 4.65e9·4-s + 2.03e12·6-s − 2.03e13·7-s + 8.30e14·8-s − 4.50e15·9-s − 1.08e17·11-s + 1.51e17·12-s − 2.32e17·13-s + 1.27e18·14-s − 1.20e19·16-s − 2.21e19·17-s + 2.82e20·18-s − 2.01e21·19-s + 6.59e20·21-s + 6.78e21·22-s + 7.09e21·23-s − 2.69e22·24-s + 1.45e22·26-s + 3.26e23·27-s + 9.46e22·28-s + 1.18e24·29-s − 9.41e23·31-s − 6.37e24·32-s + 3.51e24·33-s + 1.38e24·34-s + ⋯ |
| L(s) = 1 | − 0.676·2-s − 0.435·3-s − 0.542·4-s + 0.294·6-s − 0.230·7-s + 1.04·8-s − 0.810·9-s − 0.710·11-s + 0.236·12-s − 0.0969·13-s + 0.156·14-s − 0.163·16-s − 0.110·17-s + 0.548·18-s − 1.60·19-s + 0.100·21-s + 0.480·22-s + 0.241·23-s − 0.454·24-s + 0.0655·26-s + 0.788·27-s + 0.125·28-s + 0.876·29-s − 0.232·31-s − 0.932·32-s + 0.309·33-s + 0.0746·34-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
| \(L(17)\) |
\(\approx\) |
\(0.003135180430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003135180430\) |
| \(L(\frac{35}{2})\) |
|
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 | \( 1 + 6.26e4T + 8.58e9T^{2} \) |
| 3 | \( 1 + 3.24e7T + 5.55e15T^{2} \) |
| 7 | \( 1 + 2.03e13T + 7.73e27T^{2} \) |
| 11 | \( 1 + 1.08e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 2.32e17T + 5.75e36T^{2} \) |
| 17 | \( 1 + 2.21e19T + 4.02e40T^{2} \) |
| 19 | \( 1 + 2.01e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 7.09e21T + 8.65e44T^{2} \) |
| 29 | \( 1 - 1.18e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 9.41e23T + 1.64e49T^{2} \) |
| 37 | \( 1 + 1.19e26T + 5.63e51T^{2} \) |
| 41 | \( 1 + 5.26e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.00e27T + 8.02e53T^{2} \) |
| 47 | \( 1 + 3.06e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 2.92e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 9.11e28T + 2.74e58T^{2} \) |
| 61 | \( 1 - 2.27e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.33e30T + 1.82e60T^{2} \) |
| 71 | \( 1 + 3.44e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 3.16e29T + 3.08e61T^{2} \) |
| 79 | \( 1 + 1.55e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 5.93e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 1.27e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 1.84e32T + 3.65e65T^{2} \) |
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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
−10.96015916263052908427898441302, −10.18768943581099083668821391954, −8.868104862541210192827149305313, −8.138764684701757347200856498443, −6.70164954800597917165325422802, −5.42163335953776647578901942287, −4.42402724545576436226377875995, −2.94369618472349047299615578561, −1.56561801023263157368336375108, −0.02769273291139679522896489104,
0.02769273291139679522896489104, 1.56561801023263157368336375108, 2.94369618472349047299615578561, 4.42402724545576436226377875995, 5.42163335953776647578901942287, 6.70164954800597917165325422802, 8.138764684701757347200856498443, 8.868104862541210192827149305313, 10.18768943581099083668821391954, 10.96015916263052908427898441302
