L(s) = 1 | + 1.03i·5-s + 0.267i·7-s − 3.86·11-s + 2.46·13-s − 6.69i·17-s − 1.73i·19-s + 5.93·23-s + 3.92·25-s + 2.07i·29-s − 0.535i·31-s − 0.277·35-s + 6.46·37-s − 2.07i·41-s + 7.46i·43-s + 9.52·47-s + ⋯ |
L(s) = 1 | + 0.462i·5-s + 0.101i·7-s − 1.16·11-s + 0.683·13-s − 1.62i·17-s − 0.397i·19-s + 1.23·23-s + 0.785·25-s + 0.384i·29-s − 0.0962i·31-s − 0.0468·35-s + 1.06·37-s − 0.323i·41-s + 1.13i·43-s + 1.38·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.664210206\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.664210206\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.03iT - 5T^{2} \) |
| 7 | \( 1 - 0.267iT - 7T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 + 6.69iT - 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 5.93T + 23T^{2} \) |
| 29 | \( 1 - 2.07iT - 29T^{2} \) |
| 31 | \( 1 + 0.535iT - 31T^{2} \) |
| 37 | \( 1 - 6.46T + 37T^{2} \) |
| 41 | \( 1 + 2.07iT - 41T^{2} \) |
| 43 | \( 1 - 7.46iT - 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 + 7.45T + 59T^{2} \) |
| 61 | \( 1 - 9.39T + 61T^{2} \) |
| 67 | \( 1 + 9.73iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 9.92T + 73T^{2} \) |
| 79 | \( 1 - 15.1iT - 79T^{2} \) |
| 83 | \( 1 + 7.72T + 83T^{2} \) |
| 89 | \( 1 + 6.69iT - 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−9.240849013286114632052463319662, −8.647849207669574547154108268410, −7.52069426736546070043333932334, −7.13713524311385872332479935969, −6.08153331027514333131349925083, −5.21799325093062749756268774963, −4.46612007175428471745118419069, −3.04366525766328825292717262962, −2.61232254629017445791751252265, −0.860863024718210153044497222022,
0.952719425213923681364360251676, 2.24814175625428053578526816389, 3.42245376212811949010983934546, 4.33996389295935092616471326620, 5.30709215578723637930588097160, 5.97957743434529468043249569380, 6.96047712211576140302736192206, 7.901284392838942060188105930358, 8.506246931266579464788863271866, 9.153948574213536868641259244955