| L(s) = 1 | + 10.7·2-s + 83.9·4-s − 41.6·5-s + 0.236·7-s + 560.·8-s − 448.·10-s + 421.·11-s + 254.·13-s + 2.55·14-s + 3.34e3·16-s + 975.·17-s + 2.03e3·19-s − 3.49e3·20-s + 4.54e3·22-s − 529·23-s − 1.39e3·25-s + 2.74e3·26-s + 19.8·28-s − 2.67e3·29-s + 9.03e3·31-s + 1.80e4·32-s + 1.05e4·34-s − 9.85·35-s − 1.26e4·37-s + 2.19e4·38-s − 2.33e4·40-s − 1.01e4·41-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.62·4-s − 0.744·5-s + 0.00182·7-s + 3.09·8-s − 1.41·10-s + 1.05·11-s + 0.417·13-s + 0.00347·14-s + 3.26·16-s + 0.818·17-s + 1.29·19-s − 1.95·20-s + 2.00·22-s − 0.208·23-s − 0.445·25-s + 0.795·26-s + 0.00479·28-s − 0.589·29-s + 1.68·31-s + 3.12·32-s + 1.55·34-s − 0.00135·35-s − 1.52·37-s + 2.46·38-s − 2.30·40-s − 0.942·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.756540092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.756540092\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 - 10.7T + 32T^{2} \) |
| 5 | \( 1 + 41.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 0.236T + 1.68e4T^{2} \) |
| 11 | \( 1 - 421.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 254.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 975.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.03e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 2.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.01e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.95e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.76e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.19e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.98e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.85e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.83e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.73e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.40e5T + 8.58e9T^{2} \) |
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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
−11.83136901596434430941470193561, −11.19353850887812353827620284214, −9.848281256105182155587202461310, −8.082143968217803472809018330509, −7.07184315803980188190837272732, −6.10016702835673687079083655215, −5.00795625446597941440889590815, −3.88453505317408921205482118847, −3.18551410218446037596790780112, −1.43984990546038260019325864499,
1.43984990546038260019325864499, 3.18551410218446037596790780112, 3.88453505317408921205482118847, 5.00795625446597941440889590815, 6.10016702835673687079083655215, 7.07184315803980188190837272732, 8.082143968217803472809018330509, 9.848281256105182155587202461310, 11.19353850887812353827620284214, 11.83136901596434430941470193561
