L(s) = 1 | + (0.951 + 1.30i)2-s + (−0.951 − 0.309i)3-s + (−0.500 + 1.53i)4-s + (−0.499 − 1.53i)6-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (0.951 − 1.30i)12-s + (−0.587 + 0.190i)17-s + 1.61i·18-s + (0.5 + 1.53i)19-s + (0.363 + 0.5i)23-s + 24-s + (−0.587 − 0.809i)27-s + (0.190 + 0.587i)31-s + 0.999i·32-s + ⋯ |
L(s) = 1 | + (0.951 + 1.30i)2-s + (−0.951 − 0.309i)3-s + (−0.500 + 1.53i)4-s + (−0.499 − 1.53i)6-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (0.951 − 1.30i)12-s + (−0.587 + 0.190i)17-s + 1.61i·18-s + (0.5 + 1.53i)19-s + (0.363 + 0.5i)23-s + 24-s + (−0.587 − 0.809i)27-s + (0.190 + 0.587i)31-s + 0.999i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.347533203\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.347533203\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
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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
−9.758925077369481245649985174083, −8.479281200206534524959669445047, −7.82626339446573046880172265159, −7.04606161582715253151608513188, −6.46877208738685043164949372309, −5.70776239995003194994851274722, −5.14778057676956163731895621354, −4.30592670124311872824530189842, −3.40884176163966516618573501749, −1.63510527933066331543180380800,
0.892096803552340349464468896674, 2.26925028582572100117480712534, 3.24839790388749761311368701950, 4.26809131163049349859133479934, 4.83705870552929864618471692874, 5.51473185852872656977541752200, 6.51065258157281094273876591409, 7.26579751573340735848435728097, 8.639538831052658461198103849226, 9.619762096335287619548405915966