L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (1.53 + 0.5i)13-s + (0.309 − 0.951i)16-s + (0.363 − 0.5i)17-s − 0.999i·18-s − 1.61·26-s + (0.5 − 0.363i)29-s + i·32-s + (−0.190 + 0.587i)34-s + (0.309 + 0.951i)36-s + (0.587 + 0.190i)37-s + (−0.5 + 1.53i)41-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (1.53 + 0.5i)13-s + (0.309 − 0.951i)16-s + (0.363 − 0.5i)17-s − 0.999i·18-s − 1.61·26-s + (0.5 − 0.363i)29-s + i·32-s + (−0.190 + 0.587i)34-s + (0.309 + 0.951i)36-s + (0.587 + 0.190i)37-s + (−0.5 + 1.53i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6097516571\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6097516571\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−11.23484497350181169030966220824, −10.25391640092427965122692989174, −9.419570590992556373202107282856, −8.404792249482688628805979049705, −7.918874730200162577683677714885, −6.71670564344817266796950823836, −5.92341050515362157971284899219, −4.77313475693043116195313092183, −3.06875015261999150688202887396, −1.61333522460566084204331460652,
1.27399716263776711177466042119, 3.02160885439423441866354197822, 3.90535862606247224614709386924, 5.81910784176456719317638894415, 6.50973880711564053983261231600, 7.67234477715786868572793615888, 8.597411134646744528875752775525, 9.138063347214793890790482109716, 10.24597179086817327433599966012, 10.88931817019635962496444108485