| L(s) = 1 | − 10.4·2-s + 9·3-s + 76.7·4-s − 93.8·6-s + 21.9·7-s − 467.·8-s + 81·9-s + 121·11-s + 691.·12-s + 460.·13-s − 228.·14-s + 2.41e3·16-s + 2.31e3·17-s − 844.·18-s + 800.·19-s + 197.·21-s − 1.26e3·22-s + 4.78e3·23-s − 4.20e3·24-s − 4.80e3·26-s + 729·27-s + 1.68e3·28-s + 911.·29-s − 944.·31-s − 1.02e4·32-s + 1.08e3·33-s − 2.41e4·34-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 0.577·3-s + 2.39·4-s − 1.06·6-s + 0.168·7-s − 2.58·8-s + 0.333·9-s + 0.301·11-s + 1.38·12-s + 0.756·13-s − 0.311·14-s + 2.35·16-s + 1.94·17-s − 0.614·18-s + 0.508·19-s + 0.0975·21-s − 0.555·22-s + 1.88·23-s − 1.49·24-s − 1.39·26-s + 0.192·27-s + 0.405·28-s + 0.201·29-s − 0.176·31-s − 1.76·32-s + 0.174·33-s − 3.58·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.678484673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678484673\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 10.4T + 32T^{2} \) |
| 7 | \( 1 - 21.9T + 1.68e4T^{2} \) |
| 13 | \( 1 - 460.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.31e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 800.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 911.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 944.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.71e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.28e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.62e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.43e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.68e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.79e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.46e4T + 8.58e9T^{2} \) |
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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
−9.548334453985383136203376268761, −8.584184851402239688693044166088, −8.042898502251212155093139455146, −7.28302239248191801033483747157, −6.47045802957837115103973924894, −5.29519671774897708371426781496, −3.54180206951873345287134996163, −2.70667553379694967988282405794, −1.36460481638896279307194559772, −0.884452402298638085814471200814,
0.884452402298638085814471200814, 1.36460481638896279307194559772, 2.70667553379694967988282405794, 3.54180206951873345287134996163, 5.29519671774897708371426781496, 6.47045802957837115103973924894, 7.28302239248191801033483747157, 8.042898502251212155093139455146, 8.584184851402239688693044166088, 9.548334453985383136203376268761
