| L(s) = 1 | − 1.75·3-s + 2.38·5-s − 2.17·7-s + 0.0650·9-s − 4.68·11-s + 2.94·13-s − 4.18·15-s − 3.06·17-s − 19-s + 3.80·21-s − 4.21·23-s + 0.705·25-s + 5.13·27-s + 3.70·29-s + 2.04·31-s + 8.20·33-s − 5.19·35-s + 3.34·37-s − 5.15·39-s − 9.30·41-s − 10.2·43-s + 0.155·45-s − 8.97·47-s − 2.27·49-s + 5.36·51-s + 9.41·53-s − 11.1·55-s + ⋯ |
| L(s) = 1 | − 1.01·3-s + 1.06·5-s − 0.821·7-s + 0.0216·9-s − 1.41·11-s + 0.816·13-s − 1.07·15-s − 0.743·17-s − 0.229·19-s + 0.830·21-s − 0.878·23-s + 0.141·25-s + 0.988·27-s + 0.688·29-s + 0.367·31-s + 1.42·33-s − 0.877·35-s + 0.549·37-s − 0.825·39-s − 1.45·41-s − 1.56·43-s + 0.0231·45-s − 1.30·47-s − 0.325·49-s + 0.751·51-s + 1.29·53-s − 1.50·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8308967840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8308967840\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 - 2.38T + 5T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 11 | \( 1 + 4.68T + 11T^{2} \) |
| 13 | \( 1 - 2.94T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 + 9.30T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 8.97T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 8.84T + 61T^{2} \) |
| 67 | \( 1 - 4.30T + 67T^{2} \) |
| 71 | \( 1 + 2.97T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 83 | \( 1 - 6.57T + 83T^{2} \) |
| 89 | \( 1 + 4.01T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.354620413707617113735495709869, −7.06273993289217916498143979252, −6.38987119629189352717644924414, −6.04711104893185962908537340579, −5.32479710858614385662296684112, −4.77929397894032836828548457677, −3.58881367746782553193008328570, −2.68473952714073848650597669848, −1.87557678525550132018011347060, −0.48159496035040462391012496769,
0.48159496035040462391012496769, 1.87557678525550132018011347060, 2.68473952714073848650597669848, 3.58881367746782553193008328570, 4.77929397894032836828548457677, 5.32479710858614385662296684112, 6.04711104893185962908537340579, 6.38987119629189352717644924414, 7.06273993289217916498143979252, 8.354620413707617113735495709869
