L(s) = 1 | + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + 1.61·7-s + (−0.809 + 0.587i)9-s + (0.809 − 0.587i)12-s + (0.5 − 0.363i)13-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)19-s + (−0.500 − 1.53i)21-s + (0.809 + 0.587i)27-s + (0.500 + 1.53i)28-s + (−0.5 + 1.53i)31-s + (−0.809 − 0.587i)36-s + (0.5 − 0.363i)37-s + (−0.5 − 0.363i)39-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + 1.61·7-s + (−0.809 + 0.587i)9-s + (0.809 − 0.587i)12-s + (0.5 − 0.363i)13-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)19-s + (−0.500 − 1.53i)21-s + (0.809 + 0.587i)27-s + (0.500 + 1.53i)28-s + (−0.5 + 1.53i)31-s + (−0.809 − 0.587i)36-s + (0.5 − 0.363i)37-s + (−0.5 − 0.363i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.308580267\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.308580267\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 - 1.61T + T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + 0.618T + T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 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 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−8.934173839596498724162121184353, −8.455078607187020539832769810112, −7.75394402564211537449542735906, −7.25454556870028854006836877922, −6.39650872120693945859841261664, −5.35439208001140033193948478244, −4.62567576041827928521419279713, −3.38453225229038528363561199212, −2.32050884377082311708306903452, −1.38511667375243305869908908134,
1.27772801255455193559310051757, 2.37835647475679242833823510376, 3.89929166682971568305873069144, 4.61553116559420711455332387638, 5.41872736177761530839143779434, 5.92354405970560575858865489708, 6.96953072800551463786637039179, 8.048946051947926558318513930539, 8.707618684945763733187909925043, 9.646519537163549999959866496008