| L(s) = 1 | − 1.81·2-s − 3-s + 1.29·4-s + 5-s + 1.81·6-s − 4.23·7-s + 1.28·8-s + 9-s − 1.81·10-s − 4.07·11-s − 1.29·12-s + 5.12·13-s + 7.68·14-s − 15-s − 4.91·16-s + 6.56·17-s − 1.81·18-s − 0.774·19-s + 1.29·20-s + 4.23·21-s + 7.38·22-s − 1.28·24-s + 25-s − 9.30·26-s − 27-s − 5.47·28-s − 8.18·29-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.577·3-s + 0.647·4-s + 0.447·5-s + 0.740·6-s − 1.59·7-s + 0.452·8-s + 0.333·9-s − 0.573·10-s − 1.22·11-s − 0.373·12-s + 1.42·13-s + 2.05·14-s − 0.258·15-s − 1.22·16-s + 1.59·17-s − 0.427·18-s − 0.177·19-s + 0.289·20-s + 0.923·21-s + 1.57·22-s − 0.261·24-s + 0.200·25-s − 1.82·26-s − 0.192·27-s − 1.03·28-s − 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5374590165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5374590165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 4.07T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 6.56T + 17T^{2} \) |
| 19 | \( 1 + 0.774T + 19T^{2} \) |
| 29 | \( 1 + 8.18T + 29T^{2} \) |
| 31 | \( 1 - 4.71T + 31T^{2} \) |
| 37 | \( 1 + 5.63T + 37T^{2} \) |
| 41 | \( 1 - 7.84T + 41T^{2} \) |
| 43 | \( 1 - 2.93T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 - 8.07T + 53T^{2} \) |
| 59 | \( 1 - 6.14T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 6.12T + 71T^{2} \) |
| 73 | \( 1 + 7.10T + 73T^{2} \) |
| 79 | \( 1 + 9.02T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 8.31T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.906036464578777341188310365090, −7.21716336252399125177932019496, −6.61447094812553992320240175566, −5.63893074545970091366168249178, −5.60357721070195060542689800239, −4.18326173001845498166198215470, −3.39041996960495793511382038348, −2.51093876433478146391218198786, −1.35151729566470675512917159557, −0.49808719943551509526580692928,
0.49808719943551509526580692928, 1.35151729566470675512917159557, 2.51093876433478146391218198786, 3.39041996960495793511382038348, 4.18326173001845498166198215470, 5.60357721070195060542689800239, 5.63893074545970091366168249178, 6.61447094812553992320240175566, 7.21716336252399125177932019496, 7.906036464578777341188310365090
