L(s) = 1 | + (−0.363 + 0.5i)2-s + (−0.951 + 0.309i)3-s + (0.190 + 0.587i)4-s + (0.190 − 0.587i)6-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.363 − 0.5i)12-s + (1.53 + 0.5i)17-s + 0.618i·18-s + (−0.190 + 0.587i)19-s + (−0.951 + 1.30i)23-s + 24-s + (−0.587 + 0.809i)27-s + (−0.5 + 1.53i)31-s − i·32-s + ⋯ |
L(s) = 1 | + (−0.363 + 0.5i)2-s + (−0.951 + 0.309i)3-s + (0.190 + 0.587i)4-s + (0.190 − 0.587i)6-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.363 − 0.5i)12-s + (1.53 + 0.5i)17-s + 0.618i·18-s + (−0.190 + 0.587i)19-s + (−0.951 + 1.30i)23-s + 24-s + (−0.587 + 0.809i)27-s + (−0.5 + 1.53i)31-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5796063105\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5796063105\) |
\(L(1)\) |
|
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 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 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 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 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
−9.765224423118374344063916494187, −8.980343595175711476673770834517, −7.955980566176338082415465354711, −7.49798573020153950696922436609, −6.54086270521985518182006019775, −5.88320197339552104507289470053, −5.15259478984080704100681041780, −3.87506081573270231784031311022, −3.30442800274761379242717010385, −1.54253241556230496239728393717,
0.56209824533472429316474820630, 1.76643323881772306589158525205, 2.80245801912139242719806423166, 4.23286179303830027007996418949, 5.21896134119435286140485011761, 5.89626650961487108142941198681, 6.54335665946741205244170756590, 7.47713109370612519708798320584, 8.283657772137164058639026200591, 9.459690823378526598673320342442