L(s) = 1 | − 3.07·3-s − 3.53·5-s − 7-s + 6.42·9-s + 11-s − 13-s + 10.8·15-s + 1.83·17-s − 0.318·19-s + 3.07·21-s + 1.10·23-s + 7.53·25-s − 10.5·27-s − 1.36·29-s − 1.34·31-s − 3.07·33-s + 3.53·35-s + 10.6·37-s + 3.07·39-s + 0.852·41-s − 2.13·43-s − 22.7·45-s + 10.5·47-s + 49-s − 5.63·51-s − 1.45·53-s − 3.53·55-s + ⋯ |
L(s) = 1 | − 1.77·3-s − 1.58·5-s − 0.377·7-s + 2.14·9-s + 0.301·11-s − 0.277·13-s + 2.80·15-s + 0.445·17-s − 0.0729·19-s + 0.670·21-s + 0.230·23-s + 1.50·25-s − 2.02·27-s − 0.252·29-s − 0.241·31-s − 0.534·33-s + 0.598·35-s + 1.75·37-s + 0.491·39-s + 0.133·41-s − 0.326·43-s − 3.39·45-s + 1.53·47-s + 0.142·49-s − 0.789·51-s − 0.199·53-s − 0.477·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4336640000\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4336640000\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 + 3.53T + 5T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 + 0.318T + 19T^{2} \) |
| 23 | \( 1 - 1.10T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 + 1.34T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 0.852T + 41T^{2} \) |
| 43 | \( 1 + 2.13T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 1.45T + 53T^{2} \) |
| 59 | \( 1 + 5.50T + 59T^{2} \) |
| 61 | \( 1 + 4.79T + 61T^{2} \) |
| 67 | \( 1 - 1.55T + 67T^{2} \) |
| 71 | \( 1 - 0.673T + 71T^{2} \) |
| 73 | \( 1 - 2.83T + 73T^{2} \) |
| 79 | \( 1 + 6.33T + 79T^{2} \) |
| 83 | \( 1 + 4.51T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 3.08T + 97T^{2} \) |
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
−7.52938690042910109002251476224, −7.17617144046848386464100849437, −6.40732025799591846260330755293, −5.79184368491118776972458331621, −5.03636667417531940615898440087, −4.29575759022963626706274828599, −3.88494329760799009299835010364, −2.82522564737491086171870782116, −1.26138472878929005896051748070, −0.40772237337830975635546737568,
0.40772237337830975635546737568, 1.26138472878929005896051748070, 2.82522564737491086171870782116, 3.88494329760799009299835010364, 4.29575759022963626706274828599, 5.03636667417531940615898440087, 5.79184368491118776972458331621, 6.40732025799591846260330755293, 7.17617144046848386464100849437, 7.52938690042910109002251476224