| L(s) = 1 | − 1.88i·2-s − 1.56·4-s − 1.89·5-s − 0.823i·8-s + 3.58i·10-s + 1.25i·11-s + 6.80i·13-s − 4.68·16-s + 7.12·17-s − 7.22i·19-s + 2.96·20-s + 2.37·22-s − 3.66i·23-s − 1.39·25-s + 12.8·26-s + ⋯ |
| L(s) = 1 | − 1.33i·2-s − 0.781·4-s − 0.849·5-s − 0.291i·8-s + 1.13i·10-s + 0.379i·11-s + 1.88i·13-s − 1.17·16-s + 1.72·17-s − 1.65i·19-s + 0.663·20-s + 0.506·22-s − 0.764i·23-s − 0.278·25-s + 2.51·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6203646273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6203646273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.88iT - 2T^{2} \) |
| 5 | \( 1 + 1.89T + 5T^{2} \) |
| 11 | \( 1 - 1.25iT - 11T^{2} \) |
| 13 | \( 1 - 6.80iT - 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 + 7.22iT - 19T^{2} \) |
| 23 | \( 1 + 3.66iT - 23T^{2} \) |
| 29 | \( 1 + 6.95iT - 29T^{2} \) |
| 31 | \( 1 + 4.28iT - 31T^{2} \) |
| 37 | \( 1 + 9.49T + 37T^{2} \) |
| 41 | \( 1 - 2.61T + 41T^{2} \) |
| 43 | \( 1 - 2.15T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 - 2.36iT - 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 7.92iT - 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 8.63iT - 71T^{2} \) |
| 73 | \( 1 - 9.20iT - 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 1.57T + 89T^{2} \) |
| 97 | \( 1 + 1.36iT - 97T^{2} \) |
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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
−8.430275615389250731210771534898, −7.36129104827136489076982326511, −6.95610909201722625183839424780, −5.88814340255008064426270414754, −4.47569763224242036045225189033, −4.29934700114703584215824585946, −3.28455978786007219977352565246, −2.42943827092192883077944586767, −1.46916741561563935914648499127, −0.20113535940393525504925679262,
1.38028716447613902548521499040, 3.21767793368288933539543919640, 3.56167547158965671048559540247, 4.95939757414686703325406411583, 5.61029599441284922673778581098, 5.98724806173874435183261873038, 7.25207814335550722895476540791, 7.62063930294986524643837513223, 8.184968608871988865261154339551, 8.676909240559529311455698030428
