L(s) = 1 | − i·2-s + (−0.529 − 1.64i)3-s − 4-s − 2.93·5-s + (−1.64 + 0.529i)6-s + i·7-s + i·8-s + (−2.43 + 1.74i)9-s + 2.93i·10-s − 2.76·11-s + (0.529 + 1.64i)12-s + 4.19·13-s + 14-s + (1.55 + 4.84i)15-s + 16-s − 1.10·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.305 − 0.952i)3-s − 0.5·4-s − 1.31·5-s + (−0.673 + 0.216i)6-s + 0.377i·7-s + 0.353i·8-s + (−0.813 + 0.581i)9-s + 0.929i·10-s − 0.832·11-s + (0.152 + 0.476i)12-s + 1.16·13-s + 0.267·14-s + (0.401 + 1.25i)15-s + 0.250·16-s − 0.268·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.689299 - 0.0961685i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.689299 - 0.0961685i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.529 + 1.64i)T \) |
| 7 | \( 1 - iT \) |
| 23 | \( 1 + (-2.65 + 3.99i)T \) |
good | 5 | \( 1 + 2.93T + 5T^{2} \) |
| 11 | \( 1 + 2.76T + 11T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 - 4.25iT - 19T^{2} \) |
| 29 | \( 1 + 0.175iT - 29T^{2} \) |
| 31 | \( 1 - 3.49T + 31T^{2} \) |
| 37 | \( 1 + 1.18iT - 37T^{2} \) |
| 41 | \( 1 - 6.45iT - 41T^{2} \) |
| 43 | \( 1 - 5.95iT - 43T^{2} \) |
| 47 | \( 1 + 6.79iT - 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 - 10.5iT - 59T^{2} \) |
| 61 | \( 1 - 5.65iT - 61T^{2} \) |
| 67 | \( 1 + 7.63iT - 67T^{2} \) |
| 71 | \( 1 - 5.46iT - 71T^{2} \) |
| 73 | \( 1 + 5.68T + 73T^{2} \) |
| 79 | \( 1 - 0.0580iT - 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 7.60T + 89T^{2} \) |
| 97 | \( 1 - 1.20iT - 97T^{2} \) |
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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
−10.36476833594366682312868424645, −8.895247324968706085461151881021, −8.228662920782167957817909644735, −7.73232083902517820844692315436, −6.60211973276177275662008133850, −5.66064746326386139795579883515, −4.56704716883807985511292475260, −3.48342939207675281870008983351, −2.46060637561400786227888364138, −1.00139241827342335897270952908,
0.44081435350771085294573959889, 3.16903291693120247092603850289, 3.96501658137621310250750196510, 4.74512414784714982502101155239, 5.61723063742700297657208097646, 6.71191866090922077439053931368, 7.56663249048084997863206736969, 8.408571939827519704001029127998, 9.012459643025053533354176603690, 10.07273238232199535608315288731