| L(s) = 1 | + 7.20·2-s + 9·3-s + 19.9·4-s + 88.7·5-s + 64.8·6-s − 52.9·7-s − 86.9·8-s + 81·9-s + 639.·10-s − 129.·11-s + 179.·12-s + 525.·13-s − 381.·14-s + 798.·15-s − 1.26e3·16-s + 2.07e3·17-s + 583.·18-s + 736.·19-s + 1.76e3·20-s − 476.·21-s − 935.·22-s + 1.96e3·23-s − 782.·24-s + 4.75e3·25-s + 3.78e3·26-s + 729·27-s − 1.05e3·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.577·3-s + 0.622·4-s + 1.58·5-s + 0.735·6-s − 0.408·7-s − 0.480·8-s + 0.333·9-s + 2.02·10-s − 0.323·11-s + 0.359·12-s + 0.862·13-s − 0.520·14-s + 0.916·15-s − 1.23·16-s + 1.74·17-s + 0.424·18-s + 0.467·19-s + 0.988·20-s − 0.235·21-s − 0.412·22-s + 0.772·23-s − 0.277·24-s + 1.52·25-s + 1.09·26-s + 0.192·27-s − 0.254·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.538508487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.538508487\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 103 | \( 1 + 1.06e4T \) |
| good | 2 | \( 1 - 7.20T + 32T^{2} \) |
| 5 | \( 1 - 88.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 52.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 129.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 525.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.07e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 736.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.96e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.99e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.35e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.02e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.95e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.90e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.26e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.92e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.10e5T + 8.58e9T^{2} \) |
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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
−10.89052631525846022549308593953, −9.711222272333761098760020475356, −9.300920829438643342573199238973, −7.917203631667528619687625315014, −6.45372705829708454452436274540, −5.80046241864883960078353407237, −4.92742105082903499706687696286, −3.45837180200726486943159898885, −2.72341750337238448531225913403, −1.32273762897230519684824037435,
1.32273762897230519684824037435, 2.72341750337238448531225913403, 3.45837180200726486943159898885, 4.92742105082903499706687696286, 5.80046241864883960078353407237, 6.45372705829708454452436274540, 7.917203631667528619687625315014, 9.300920829438643342573199238973, 9.711222272333761098760020475356, 10.89052631525846022549308593953
