L(s) = 1 | + 1.80·2-s − 0.445·3-s + 2.24·4-s + 5-s − 0.801·6-s + 2.24·8-s − 0.801·9-s + 1.80·10-s − 11-s − 12-s − 1.24·13-s − 0.445·15-s + 1.80·16-s − 17-s − 1.44·18-s + 2.24·20-s − 1.80·22-s − 1.80·23-s − 1.00·24-s + 25-s − 2.24·26-s + 0.801·27-s + 1.80·29-s − 0.801·30-s + 1.00·32-s + 0.445·33-s − 1.80·34-s + ⋯ |
L(s) = 1 | + 1.80·2-s − 0.445·3-s + 2.24·4-s + 5-s − 0.801·6-s + 2.24·8-s − 0.801·9-s + 1.80·10-s − 11-s − 12-s − 1.24·13-s − 0.445·15-s + 1.80·16-s − 17-s − 1.44·18-s + 2.24·20-s − 1.80·22-s − 1.80·23-s − 1.00·24-s + 25-s − 2.24·26-s + 0.801·27-s + 1.80·29-s − 0.801·30-s + 1.00·32-s + 0.445·33-s − 1.80·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.421888952\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.421888952\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - 1.80T + T^{2} \) |
| 3 | \( 1 + 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.24T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 - 1.80T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.24T + T^{2} \) |
| 41 | \( 1 - 0.445T + T^{2} \) |
| 43 | \( 1 - 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.24T + T^{2} \) |
| 61 | \( 1 + 1.24T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 0.445T + T^{2} \) |
| 83 | \( 1 + 1.24T + T^{2} \) |
| 89 | \( 1 + 0.445T + T^{2} \) |
| 97 | \( 1 + 1.80T + T^{2} \) |
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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
−10.52496179434494908931088473080, −9.801691970085044940789206668151, −8.439219859351225014120989268688, −7.30912329500406299507646323438, −6.31843337337925377973654757861, −5.81997867083275694886673792111, −5.02979630405860432621925846848, −4.35035911343563258485772232678, −2.70825798849540986994212113545, −2.36056177126232578873588065134,
2.36056177126232578873588065134, 2.70825798849540986994212113545, 4.35035911343563258485772232678, 5.02979630405860432621925846848, 5.81997867083275694886673792111, 6.31843337337925377973654757861, 7.30912329500406299507646323438, 8.439219859351225014120989268688, 9.801691970085044940789206668151, 10.52496179434494908931088473080