| L(s) = 1 | + 9.88·3-s + 18.7·7-s + 70.7·9-s + 16.8·11-s + 92.5·13-s + 49.4·17-s + 59.3·19-s + 185.·21-s − 44.8·23-s + 432.·27-s − 6.94·29-s − 110.·31-s + 166.·33-s − 222.·37-s + 914.·39-s − 341.·41-s − 178.·43-s + 415.·47-s + 8.86·49-s + 488.·51-s − 298.·53-s + 586.·57-s − 471.·59-s − 509.·61-s + 1.32e3·63-s + 61.1·67-s − 443.·69-s + ⋯ |
| L(s) = 1 | + 1.90·3-s + 1.01·7-s + 2.61·9-s + 0.461·11-s + 1.97·13-s + 0.704·17-s + 0.716·19-s + 1.92·21-s − 0.406·23-s + 3.07·27-s − 0.0444·29-s − 0.642·31-s + 0.877·33-s − 0.987·37-s + 3.75·39-s − 1.30·41-s − 0.634·43-s + 1.28·47-s + 0.0258·49-s + 1.34·51-s − 0.774·53-s + 1.36·57-s − 1.04·59-s − 1.07·61-s + 2.65·63-s + 0.111·67-s − 0.773·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.842550904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.842550904\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 9.88T + 27T^{2} \) |
| 7 | \( 1 - 18.7T + 343T^{2} \) |
| 11 | \( 1 - 16.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 92.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 49.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 44.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6.94T + 2.43e4T^{2} \) |
| 31 | \( 1 + 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 222.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 341.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 178.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 415.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 298.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 471.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 509.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 61.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 492.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 530.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 365.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 746.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 323.T + 9.12e5T^{2} \) |
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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
−8.697017614659886830953838791910, −8.172151267633509889211638553727, −7.57233402160364561938506102894, −6.69613694978007549578982460625, −5.55054703362987885644438863575, −4.40269993036866028971848556490, −3.64073351348712681851675798269, −3.07353548458572352286244598287, −1.65691296860350364193947345936, −1.38489668928462197953907362536,
1.38489668928462197953907362536, 1.65691296860350364193947345936, 3.07353548458572352286244598287, 3.64073351348712681851675798269, 4.40269993036866028971848556490, 5.55054703362987885644438863575, 6.69613694978007549578982460625, 7.57233402160364561938506102894, 8.172151267633509889211638553727, 8.697017614659886830953838791910
