| L(s) = 1 | + 2.76i·5-s − i·7-s + 2.55·11-s − 2.32·13-s + 7.36i·17-s − 1.78i·19-s + 3.87·23-s − 2.64·25-s + 7.32i·29-s − 2.17i·31-s + 2.76·35-s − 1.24·37-s − 3.52i·41-s + 5.28i·43-s − 5.77·47-s + ⋯ |
| L(s) = 1 | + 1.23i·5-s − 0.377i·7-s + 0.770·11-s − 0.645·13-s + 1.78i·17-s − 0.408i·19-s + 0.807·23-s − 0.529·25-s + 1.36i·29-s − 0.390i·31-s + 0.467·35-s − 0.205·37-s − 0.550i·41-s + 0.806i·43-s − 0.842·47-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
| \(L(1)\) |
\(\approx\) |
\(1.473355167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473355167\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.76iT - 5T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 + 2.32T + 13T^{2} \) |
| 17 | \( 1 - 7.36iT - 17T^{2} \) |
| 19 | \( 1 + 1.78iT - 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 - 7.32iT - 29T^{2} \) |
| 31 | \( 1 + 2.17iT - 31T^{2} \) |
| 37 | \( 1 + 1.24T + 37T^{2} \) |
| 41 | \( 1 + 3.52iT - 41T^{2} \) |
| 43 | \( 1 - 5.28iT - 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 + 7.29iT - 53T^{2} \) |
| 59 | \( 1 - 4.98T + 59T^{2} \) |
| 61 | \( 1 - 3.15T + 61T^{2} \) |
| 67 | \( 1 - 7.72iT - 67T^{2} \) |
| 71 | \( 1 + 7.98T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 2.10iT - 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 9.71iT - 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.302007402971296795057867672324, −7.46628784067176722633053997143, −6.81697221364296891830278266363, −6.49732160827761058043486998811, −5.57611746918071937374418372088, −4.65481378066110623467940635735, −3.77010474189666573483452066561, −3.22153307725462746771090516985, −2.26246277873010269252054432574, −1.27236023300952909742567700878,
0.39363081373848257151058214323, 1.34103497298018136432444066605, 2.40302862378040996551745235890, 3.30468126089863249190805886395, 4.40764672084674339417854233368, 4.87498923368827570469913965656, 5.50815597343085161573031259003, 6.36661504302818628887099215468, 7.18828685763309211467349620947, 7.80985473980915247857362312384
