L(s) = 1 | + (−1.93 − 0.306i)2-s + (2.68 + 0.873i)4-s + (0.891 − 0.453i)7-s + (−3.18 − 1.62i)8-s + (−0.587 − 0.809i)9-s + (−0.104 + 0.994i)11-s + (−1.86 + 0.604i)14-s + (3.36 + 2.44i)16-s + (0.888 + 1.74i)18-s + (0.506 − 1.88i)22-s + (1.05 + 1.05i)23-s + (2.79 − 0.442i)28-s + (0.614 − 1.89i)29-s + (−3.23 − 3.23i)32-s + (−0.873 − 2.68i)36-s + (0.188 + 0.370i)37-s + ⋯ |
L(s) = 1 | + (−1.93 − 0.306i)2-s + (2.68 + 0.873i)4-s + (0.891 − 0.453i)7-s + (−3.18 − 1.62i)8-s + (−0.587 − 0.809i)9-s + (−0.104 + 0.994i)11-s + (−1.86 + 0.604i)14-s + (3.36 + 2.44i)16-s + (0.888 + 1.74i)18-s + (0.506 − 1.88i)22-s + (1.05 + 1.05i)23-s + (2.79 − 0.442i)28-s + (0.614 − 1.89i)29-s + (−3.23 − 3.23i)32-s + (−0.873 − 2.68i)36-s + (0.188 + 0.370i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5220407735\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5220407735\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (-0.891 + 0.453i)T \) |
| 11 | \( 1 + (0.104 - 0.994i)T \) |
good | 2 | \( 1 + (1.93 + 0.306i)T + (0.951 + 0.309i)T^{2} \) |
| 3 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 13 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-1.05 - 1.05i)T + iT^{2} \) |
| 29 | \( 1 + (-0.614 + 1.89i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.188 - 0.370i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (-0.147 + 0.147i)T - iT^{2} \) |
| 47 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (-1.16 - 0.183i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.575 + 0.575i)T - iT^{2} \) |
| 71 | \( 1 + (-0.169 - 0.122i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (0.658 - 0.478i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.951 + 0.309i)T^{2} \) |
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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
−9.389285810487554739836003830152, −8.566684174810928365242601737328, −7.939928814906734398086077841573, −7.26023596745920293506708916834, −6.61428960471972090039293608735, −5.56905554502002009310761516019, −4.14616180920938185669481112820, −2.96767500069347963748350711297, −1.98947619742273854630200654725, −0.905709021936651589748937021582,
1.07248431948081732684359723094, 2.25983905609236303356703071087, 3.02159330559130259540118036768, 5.03220233036835373053657615227, 5.68543818577872876633800199719, 6.60524645588978139905868687246, 7.43146483305971946117570066301, 8.177668288644548214366540451500, 8.733610071645157570130355047357, 9.012944404946030065034146405555