| L(s) = 1 | + 50.0·5-s + 49·7-s + 150.·11-s + 522.·13-s − 345.·17-s + 2.14e3·19-s − 484.·23-s − 618.·25-s + 3.94e3·29-s + 3.90e3·31-s + 2.45e3·35-s − 5.57e3·37-s + 7.70e3·41-s − 2.56e3·43-s + 2.09e4·47-s + 2.40e3·49-s − 2.96e4·53-s + 7.53e3·55-s + 1.48e4·59-s + 4.98e3·61-s + 2.61e4·65-s − 3.59e4·67-s + 1.31e4·71-s + 4.91e4·73-s + 7.37e3·77-s + 1.72e4·79-s − 1.79e4·83-s + ⋯ |
| L(s) = 1 | + 0.895·5-s + 0.377·7-s + 0.374·11-s + 0.857·13-s − 0.289·17-s + 1.36·19-s − 0.191·23-s − 0.197·25-s + 0.871·29-s + 0.729·31-s + 0.338·35-s − 0.669·37-s + 0.715·41-s − 0.211·43-s + 1.38·47-s + 0.142·49-s − 1.45·53-s + 0.335·55-s + 0.556·59-s + 0.171·61-s + 0.767·65-s − 0.977·67-s + 0.310·71-s + 1.07·73-s + 0.141·77-s + 0.311·79-s − 0.285·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.507870198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.507870198\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 5 | \( 1 - 50.0T + 3.12e3T^{2} \) |
| 11 | \( 1 - 150.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 522.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 345.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.14e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 484.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.57e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.70e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.56e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.96e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.48e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.98e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.79e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.47e4T + 8.58e9T^{2} \) |
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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
−9.279505440311409510940030928037, −8.492119002216710172693424761220, −7.58644206470042750013546454121, −6.55628257758040627552209781223, −5.85434356974668533680820961808, −5.00374730944413479346763454658, −3.93076841298270700560895154842, −2.82220296945142636121080937299, −1.71116122489511082502060352037, −0.871090507384922915339981587272,
0.871090507384922915339981587272, 1.71116122489511082502060352037, 2.82220296945142636121080937299, 3.93076841298270700560895154842, 5.00374730944413479346763454658, 5.85434356974668533680820961808, 6.55628257758040627552209781223, 7.58644206470042750013546454121, 8.492119002216710172693424761220, 9.279505440311409510940030928037
