| L(s) = 1 | + 22.8·3-s + 105.·5-s − 182.·7-s + 277.·9-s + 419.·11-s + 494.·13-s + 2.41e3·15-s + 289.·17-s − 284.·19-s − 4.17e3·21-s + 2.40e3·23-s + 8.03e3·25-s + 797.·27-s − 2.81e3·29-s + 935.·31-s + 9.57e3·33-s − 1.93e4·35-s + 7.02e3·37-s + 1.12e4·39-s − 1.68e3·41-s − 2.03e4·43-s + 2.93e4·45-s − 1.93e4·47-s + 1.66e4·49-s + 6.59e3·51-s + 6.89e3·53-s + 4.43e4·55-s + ⋯ |
| L(s) = 1 | + 1.46·3-s + 1.89·5-s − 1.41·7-s + 1.14·9-s + 1.04·11-s + 0.811·13-s + 2.76·15-s + 0.242·17-s − 0.180·19-s − 2.06·21-s + 0.948·23-s + 2.57·25-s + 0.210·27-s − 0.620·29-s + 0.174·31-s + 1.52·33-s − 2.66·35-s + 0.843·37-s + 1.18·39-s − 0.156·41-s − 1.68·43-s + 2.16·45-s − 1.27·47-s + 0.990·49-s + 0.355·51-s + 0.337·53-s + 1.97·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.665321351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.665321351\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + 1.68e3T \) |
| good | 3 | \( 1 - 22.8T + 243T^{2} \) |
| 5 | \( 1 - 105.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 182.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 419.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 494.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 289.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 284.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.40e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.81e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 935.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.02e3T + 6.93e7T^{2} \) |
| 43 | \( 1 + 2.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.93e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.89e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.93e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.35e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.79e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.11e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.75e4T + 8.58e9T^{2} \) |
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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
−9.732594343618921711308961747854, −9.040397060580985195990021263223, −8.426150972844399296423591960153, −6.83258569860525605919710357152, −6.43966274350986336254708237723, −5.39384486133036992609981808748, −3.76663499936728674645244199326, −3.03618362130840142532287516435, −2.11795741834219363767235348896, −1.14579047637928078122148431320,
1.14579047637928078122148431320, 2.11795741834219363767235348896, 3.03618362130840142532287516435, 3.76663499936728674645244199326, 5.39384486133036992609981808748, 6.43966274350986336254708237723, 6.83258569860525605919710357152, 8.426150972844399296423591960153, 9.040397060580985195990021263223, 9.732594343618921711308961747854
