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
−7.80985473980915247857362312384, −7.18828685763309211467349620947, −6.36661504302818628887099215468, −5.50815597343085161573031259003, −4.87498923368827570469913965656, −4.40764672084674339417854233368, −3.30468126089863249190805886395, −2.40302862378040996551745235890, −1.34103497298018136432444066605, −0.39363081373848257151058214323,
1.27236023300952909742567700878, 2.26246277873010269252054432574, 3.22153307725462746771090516985, 3.77010474189666573483452066561, 4.65481378066110623467940635735, 5.57611746918071937374418372088, 6.49732160827761058043486998811, 6.81697221364296891830278266363, 7.46628784067176722633053997143, 8.302007402971296795057867672324