| L(s) = 1 | + 3.14i·5-s − i·7-s − 2.04·11-s + 2.44·13-s + 3.14i·17-s − 4.44i·19-s − 2.82·23-s − 4.89·25-s + 0.778i·29-s − 0.449i·31-s + 3.14·35-s − 7·37-s + 2.51i·41-s − 4.34i·43-s − 7.38·47-s + ⋯ |
| L(s) = 1 | + 1.40i·5-s − 0.377i·7-s − 0.618·11-s + 0.679·13-s + 0.763i·17-s − 1.02i·19-s − 0.589·23-s − 0.979·25-s + 0.144i·29-s − 0.0807i·31-s + 0.531·35-s − 1.15·37-s + 0.392i·41-s − 0.663i·43-s − 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03442758122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03442758122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 3.14iT - 5T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 3.14iT - 17T^{2} \) |
| 19 | \( 1 + 4.44iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 0.778iT - 29T^{2} \) |
| 31 | \( 1 + 0.449iT - 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 - 2.51iT - 41T^{2} \) |
| 43 | \( 1 + 4.34iT - 43T^{2} \) |
| 47 | \( 1 + 7.38T + 47T^{2} \) |
| 53 | \( 1 - 6.29iT - 53T^{2} \) |
| 59 | \( 1 + 5.83T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 13.4iT - 79T^{2} \) |
| 83 | \( 1 + 6.75T + 83T^{2} \) |
| 89 | \( 1 + 14.1iT - 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.76173701607546856814078215970, −6.99626469508748327688592224271, −6.52496324582115304427700126135, −5.83387853673663178296426297709, −4.92497757022656492821225302072, −3.96609296999329002912865494468, −3.28532678419523672433887537579, −2.57881324194787748312886762594, −1.58037579211822471829780583338, −0.008677174784077116089984647120,
1.20554259665473607627727570593, 2.03017784336611726609759652918, 3.17686898089117780141469024172, 4.03060166238075920396756836548, 4.87320404865296229692666548547, 5.38875000476502379798603979293, 6.03494979251750237689019184688, 6.91952524226457725158985374480, 8.030869966620906025866996844101, 8.172714548591770328806313699365
