| L(s) = 1 | − 4.04·2-s + 8.39·4-s − 7·7-s − 1.58·8-s − 6.78·11-s − 48.9·13-s + 28.3·14-s − 60.7·16-s + 92.4·17-s − 125.·19-s + 27.4·22-s − 32.2·23-s + 198.·26-s − 58.7·28-s − 282.·29-s + 205.·31-s + 258.·32-s − 374.·34-s − 190.·37-s + 508.·38-s − 123.·41-s − 35.0·43-s − 56.9·44-s + 130.·46-s + 419.·47-s + 49·49-s − 410.·52-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.04·4-s − 0.377·7-s − 0.0698·8-s − 0.185·11-s − 1.04·13-s + 0.541·14-s − 0.948·16-s + 1.31·17-s − 1.51·19-s + 0.266·22-s − 0.292·23-s + 1.49·26-s − 0.396·28-s − 1.81·29-s + 1.19·31-s + 1.42·32-s − 1.88·34-s − 0.846·37-s + 2.17·38-s − 0.469·41-s − 0.124·43-s − 0.194·44-s + 0.418·46-s + 1.30·47-s + 0.142·49-s − 1.09·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4523708764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4523708764\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 4.04T + 8T^{2} \) |
| 11 | \( 1 + 6.78T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 92.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 32.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 282.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 123.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 35.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 419.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 0.365T + 1.48e5T^{2} \) |
| 59 | \( 1 + 328.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 515.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 828.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 496.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 701.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 199.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 194.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 137.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 220.T + 9.12e5T^{2} \) |
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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
−9.126969564468671162817805354610, −8.312579830300902485976914638964, −7.64646020781857451799582637977, −6.99372952853912235215307872227, −6.03968302323195183925845356716, −4.99163236454800849955538819013, −3.89582506953661236301918246383, −2.61668636140846254216483382482, −1.68628712015037469589873447939, −0.40020706403351409556388693105,
0.40020706403351409556388693105, 1.68628712015037469589873447939, 2.61668636140846254216483382482, 3.89582506953661236301918246383, 4.99163236454800849955538819013, 6.03968302323195183925845356716, 6.99372952853912235215307872227, 7.64646020781857451799582637977, 8.312579830300902485976914638964, 9.126969564468671162817805354610
