| L(s) = 1 | + 3.46·5-s + (−1 − 2.44i)7-s + 4.24i·11-s + 4.89i·13-s − 3.46·17-s − 4.89i·19-s + 4.24i·23-s + 6.99·25-s + 4.24i·29-s + (−3.46 − 8.48i)35-s + 8·37-s + 3.46·41-s + 2·43-s + 6.92·47-s + (−4.99 + 4.89i)49-s + ⋯ |
| L(s) = 1 | + 1.54·5-s + (−0.377 − 0.925i)7-s + 1.27i·11-s + 1.35i·13-s − 0.840·17-s − 1.12i·19-s + 0.884i·23-s + 1.39·25-s + 0.787i·29-s + (−0.585 − 1.43i)35-s + 1.31·37-s + 0.541·41-s + 0.304·43-s + 1.01·47-s + (−0.714 + 0.699i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.228263309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.228263309\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 2.44i)T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 4.24iT - 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 9.79iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 4.24iT - 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−9.044048817907020605099376360553, −7.47614803345134029291320060566, −7.08470871305039394242405925301, −6.41573838248703024802912973980, −5.72697217453439037680699605603, −4.54207409742788470871929897061, −4.35813734184810242931397487068, −2.86970297117786531951481282595, −2.06633295405376184393002007633, −1.26073082200919827032678816460,
0.63082445344950819065764724194, 2.00318285639502746307916065769, 2.68001446861296984545536250745, 3.45106563058061676827104829594, 4.77299667757332560752612857223, 5.65707659306126034958585900375, 6.06509946174346272272925750824, 6.38936140255097362184262552094, 7.80730156048928992156581575372, 8.389734972660595740500171301825
