L(s) = 1 | + 1.84i·2-s − 1.69i·3-s − 1.39·4-s + (1.66 − 1.49i)5-s + 3.12·6-s − 1.54i·7-s + 1.11i·8-s + 0.120·9-s + (2.75 + 3.05i)10-s + 2.53·11-s + 2.36i·12-s + 6.36i·13-s + 2.83·14-s + (−2.54 − 2.81i)15-s − 4.84·16-s + 3.54i·17-s + ⋯ |
L(s) = 1 | + 1.30i·2-s − 0.979i·3-s − 0.696·4-s + (0.742 − 0.669i)5-s + 1.27·6-s − 0.582i·7-s + 0.395i·8-s + 0.0402·9-s + (0.872 + 0.967i)10-s + 0.764·11-s + 0.682i·12-s + 1.76i·13-s + 0.758·14-s + (−0.656 − 0.727i)15-s − 1.21·16-s + 0.859i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.137138805\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.137138805\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.66 + 1.49i)T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 1.84iT - 2T^{2} \) |
| 3 | \( 1 + 1.69iT - 3T^{2} \) |
| 7 | \( 1 + 1.54iT - 7T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 13 | \( 1 - 6.36iT - 13T^{2} \) |
| 17 | \( 1 - 3.54iT - 17T^{2} \) |
| 19 | \( 1 + 0.713T + 19T^{2} \) |
| 23 | \( 1 - 1.66iT - 23T^{2} \) |
| 29 | \( 1 - 8.23T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 0.496iT - 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 - 1.50iT - 43T^{2} \) |
| 47 | \( 1 + 6.94iT - 47T^{2} \) |
| 53 | \( 1 + 9.87iT - 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 9.13T + 61T^{2} \) |
| 67 | \( 1 - 1.85iT - 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 + 8.88iT - 73T^{2} \) |
| 79 | \( 1 - 6.12T + 79T^{2} \) |
| 83 | \( 1 + 0.835iT - 83T^{2} \) |
| 89 | \( 1 + 2.27T + 89T^{2} \) |
| 97 | \( 1 - 7.56iT - 97T^{2} \) |
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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
−9.578065008007905568498947138345, −8.599961322386130422384120738688, −8.202636679085140171560854269599, −7.00501187032476253943831094800, −6.61806322439827363777532289945, −6.14235947630945136377658789120, −4.85814194440365690532585923859, −4.16939498538761217646770198890, −2.13491463761540809783931727611, −1.26579480955164131611463823727,
1.16538185645867427812578671919, 2.72194411191581178828124957229, 3.02601349234330564814297131788, 4.23196276466849189618381349801, 5.13307210976132576910305419366, 6.17922811174685905224342143691, 7.04869897493790197191366770275, 8.423524835308421305457421874134, 9.311020851196393154328194622074, 9.983956715181401987883678359626