| L(s) = 1 | + 0.272i·3-s + i·5-s − 4.61·7-s + 2.92·9-s − 2.76i·11-s − 4.63i·13-s − 0.272·15-s − 17-s + 5.52i·19-s − 1.25i·21-s + 5.47·23-s − 25-s + 1.61i·27-s + 3.06i·29-s − 9.16·31-s + ⋯ |
| L(s) = 1 | + 0.157i·3-s + 0.447i·5-s − 1.74·7-s + 0.975·9-s − 0.832i·11-s − 1.28i·13-s − 0.0704·15-s − 0.242·17-s + 1.26i·19-s − 0.274i·21-s + 1.14·23-s − 0.200·25-s + 0.311i·27-s + 0.569i·29-s − 1.64·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7087003430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7087003430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.272iT - 3T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 2.76iT - 11T^{2} \) |
| 13 | \( 1 + 4.63iT - 13T^{2} \) |
| 19 | \( 1 - 5.52iT - 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 - 3.06iT - 29T^{2} \) |
| 31 | \( 1 + 9.16T + 31T^{2} \) |
| 37 | \( 1 - 10.4iT - 37T^{2} \) |
| 41 | \( 1 - 3.77T + 41T^{2} \) |
| 43 | \( 1 - 0.00838iT - 43T^{2} \) |
| 47 | \( 1 + 0.465T + 47T^{2} \) |
| 53 | \( 1 + 2.67iT - 53T^{2} \) |
| 59 | \( 1 - 13.4iT - 59T^{2} \) |
| 61 | \( 1 - 5.47iT - 61T^{2} \) |
| 67 | \( 1 + 1.92iT - 67T^{2} \) |
| 71 | \( 1 + 6.91T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 1.11T + 79T^{2} \) |
| 83 | \( 1 - 2.75iT - 83T^{2} \) |
| 89 | \( 1 + 9.24T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−9.216141767123427348881005068267, −8.413413141047714647209120753776, −7.38569741152092148791379451532, −6.89123630237500222461261896286, −6.00591701227140118733057878053, −5.48244544744195221305066338743, −4.14161202051723879968206923672, −3.30578666524402495846246129419, −2.90687623836644222173031311096, −1.21967669062230507454487277303,
0.24275833280837321180423859382, 1.71651364124186670597227748066, 2.71572905571794406837996444003, 3.89980593167803256372732904216, 4.43678394735152867436897964599, 5.46583786284844333975229425123, 6.54101116889101783827909795962, 6.98529945233641535273511706933, 7.46822163484255831443451412113, 8.953027157463845181073568211408
