L(s) = 1 | + (0.777 − 1.54i)3-s + 3.40i·5-s + (−1.79 − 2.40i)9-s + 3.94·13-s + (5.26 + 2.64i)15-s − 6.09i·17-s + 4.38i·19-s + 8.33·23-s − 6.58·25-s + (−5.11 + 0.901i)27-s − 3.80i·29-s + 4.89i·31-s − 7.58·37-s + (3.06 − 6.10i)39-s + 0.710i·41-s + ⋯ |
L(s) = 1 | + (0.448 − 0.893i)3-s + 1.52i·5-s + (−0.597 − 0.802i)9-s + 1.09·13-s + (1.36 + 0.683i)15-s − 1.47i·17-s + 1.00i·19-s + 1.73·23-s − 1.31·25-s + (−0.984 + 0.173i)27-s − 0.707i·29-s + 0.879i·31-s − 1.24·37-s + (0.491 − 0.978i)39-s + 0.110i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.196865831\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.196865831\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.777 + 1.54i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.40iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 + 6.09iT - 17T^{2} \) |
| 19 | \( 1 - 4.38iT - 19T^{2} \) |
| 23 | \( 1 - 8.33T + 23T^{2} \) |
| 29 | \( 1 + 3.80iT - 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 + 7.58T + 37T^{2} \) |
| 41 | \( 1 - 0.710iT - 41T^{2} \) |
| 43 | \( 1 - 9.66iT - 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 1.00iT - 53T^{2} \) |
| 59 | \( 1 - 4.66T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 6.59T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 6.09iT - 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−8.915020526996671595057735228354, −8.082698287736334683426096518475, −7.29428501842082560694488124275, −6.80897967659725086771244741467, −6.19073022886682181482033307874, −5.22497615651142982016049568103, −3.71255370833893814805062186162, −3.09963897467688643330563869286, −2.34863916346160983455493667225, −1.05266019353768213869728442869,
0.900020064271311023397419323673, 2.11992196419696910433883540208, 3.51284144407523929894149377649, 4.08543161808776288631345972313, 5.05957634105159861689712346813, 5.45186912845435539189902717596, 6.57804882388653714269515183296, 7.74409120573568090373776108939, 8.624830237326801319491791287571, 8.862500830259485860516727279305