L(s) = 1 | + (0.739 − 0.673i)3-s + (−0.982 − 0.183i)4-s + (−0.404 − 1.42i)7-s + (0.0922 − 0.995i)9-s + (−0.850 + 0.526i)12-s + (0.538 + 1.89i)13-s + (0.932 + 0.361i)16-s + (0.658 + 0.600i)19-s + (−1.25 − 0.778i)21-s + (−0.602 + 0.798i)25-s + (−0.602 − 0.798i)27-s + (0.136 + 1.47i)28-s + (−1.58 − 0.614i)31-s + (−0.273 + 0.961i)36-s + (0.465 − 0.288i)37-s + ⋯ |
L(s) = 1 | + (0.739 − 0.673i)3-s + (−0.982 − 0.183i)4-s + (−0.404 − 1.42i)7-s + (0.0922 − 0.995i)9-s + (−0.850 + 0.526i)12-s + (0.538 + 1.89i)13-s + (0.932 + 0.361i)16-s + (0.658 + 0.600i)19-s + (−1.25 − 0.778i)21-s + (−0.602 + 0.798i)25-s + (−0.602 − 0.798i)27-s + (0.136 + 1.47i)28-s + (−1.58 − 0.614i)31-s + (−0.273 + 0.961i)36-s + (0.465 − 0.288i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7875806856\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7875806856\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.739 + 0.673i)T \) |
| 103 | \( 1 + (0.850 - 0.526i)T \) |
good | 2 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 5 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 7 | \( 1 + (0.404 + 1.42i)T + (-0.850 + 0.526i)T^{2} \) |
| 11 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 13 | \( 1 + (-0.538 - 1.89i)T + (-0.850 + 0.526i)T^{2} \) |
| 17 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 19 | \( 1 + (-0.658 - 0.600i)T + (0.0922 + 0.995i)T^{2} \) |
| 23 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 29 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 31 | \( 1 + (1.58 + 0.614i)T + (0.739 + 0.673i)T^{2} \) |
| 37 | \( 1 + (-0.465 + 0.288i)T + (0.445 - 0.895i)T^{2} \) |
| 41 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 43 | \( 1 + (-1.02 - 0.634i)T + (0.445 + 0.895i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 59 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 61 | \( 1 + (1.12 - 1.48i)T + (-0.273 - 0.961i)T^{2} \) |
| 67 | \( 1 + (0.0505 - 0.177i)T + (-0.850 - 0.526i)T^{2} \) |
| 71 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 73 | \( 1 + (-0.831 + 1.66i)T + (-0.602 - 0.798i)T^{2} \) |
| 79 | \( 1 + (0.243 + 0.489i)T + (-0.602 + 0.798i)T^{2} \) |
| 83 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 89 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 97 | \( 1 + (-0.726 - 0.961i)T + (-0.273 + 0.961i)T^{2} \) |
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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
−11.90205293946413255757338783735, −10.74523262210507026964211158023, −9.464484728230653354779439448570, −9.206819302172262542043182912956, −7.84649066118196175217544528737, −7.13148316670685032788513440796, −5.99508416995555671361693453387, −4.22758766611847168518148531894, −3.64252366939567837419407403303, −1.47116311445502777353997372343,
2.74873780200376509986631074823, 3.64790485077002313220877868367, 5.10750870487869113079581839819, 5.75845559360256906075967844815, 7.73500131443855944892512441097, 8.505647762336045904972405390635, 9.195301172129841668029023278072, 9.933188261336636290045122135672, 10.95140664161685825032121579389, 12.38218220347235528074810917281