L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.190 − 0.587i)3-s + (0.309 + 0.951i)4-s + (0.190 − 0.587i)6-s + 1.61·7-s + (−0.309 + 0.951i)8-s + (0.5 − 0.363i)9-s + (0.5 − 0.363i)12-s + (1.30 + 0.951i)14-s + (−0.809 + 0.587i)16-s + 0.618·18-s + (−0.309 − 0.951i)21-s + (−1.30 − 0.951i)23-s + 0.618·24-s + (−0.809 − 0.587i)27-s + (0.500 + 1.53i)28-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.190 − 0.587i)3-s + (0.309 + 0.951i)4-s + (0.190 − 0.587i)6-s + 1.61·7-s + (−0.309 + 0.951i)8-s + (0.5 − 0.363i)9-s + (0.5 − 0.363i)12-s + (1.30 + 0.951i)14-s + (−0.809 + 0.587i)16-s + 0.618·18-s + (−0.309 − 0.951i)21-s + (−1.30 − 0.951i)23-s + 0.618·24-s + (−0.809 − 0.587i)27-s + (0.500 + 1.53i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.147960916\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.147960916\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - 1.61T + T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + 0.618T + T^{2} \) |
| 47 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 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
−8.675456862175037069045740765547, −8.244183709751240449737912237485, −7.49657465986905590402014787957, −6.86836906503508669357572922047, −6.07971831909751252502381221942, −5.24759176283247482483942251059, −4.50343880762503994557043533552, −3.81165193560217516876460988218, −2.41618162202106074767020392860, −1.50764904966784089992766070636,
1.50566936130216917692412130321, 2.19302643562404637607102922316, 3.62073683677735157437505673779, 4.30111191556061734465061944626, 5.00487188715650907674742067289, 5.50004580793099174433454266777, 6.53205321172626881922714777426, 7.61070705460965652508147485744, 8.162235981269818866783288575809, 9.336404687972026935932096372607