L(s) = 1 | + 2-s − 3.24·3-s + 4-s + 3.60·5-s − 3.24·6-s − 7-s + 8-s + 7.54·9-s + 3.60·10-s − 3.40·11-s − 3.24·12-s − 14-s − 11.7·15-s + 16-s − 3.80·17-s + 7.54·18-s − 6.26·19-s + 3.60·20-s + 3.24·21-s − 3.40·22-s − 1.50·23-s − 3.24·24-s + 7.98·25-s − 14.7·27-s − 28-s − 2.49·29-s − 11.7·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.87·3-s + 0.5·4-s + 1.61·5-s − 1.32·6-s − 0.377·7-s + 0.353·8-s + 2.51·9-s + 1.13·10-s − 1.02·11-s − 0.937·12-s − 0.267·14-s − 3.02·15-s + 0.250·16-s − 0.922·17-s + 1.77·18-s − 1.43·19-s + 0.805·20-s + 0.708·21-s − 0.726·22-s − 0.314·23-s − 0.662·24-s + 1.59·25-s − 2.83·27-s − 0.188·28-s − 0.463·29-s − 2.13·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 + 6.26T + 19T^{2} \) |
| 23 | \( 1 + 1.50T + 23T^{2} \) |
| 29 | \( 1 + 2.49T + 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 - 0.493T + 37T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 - 0.347T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 7.60T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 7.87T + 61T^{2} \) |
| 67 | \( 1 - 5.07T + 67T^{2} \) |
| 71 | \( 1 + 8.31T + 71T^{2} \) |
| 73 | \( 1 + 3.42T + 73T^{2} \) |
| 79 | \( 1 + 3.06T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 0.655T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−8.721918445741363563832212642809, −7.28852276793829001896923781277, −6.64494291065885444291245960204, −6.00832734024824466460998077874, −5.59077284795149942167951913297, −4.90726479932843520709803708219, −4.08905775562965259187456852663, −2.47072324790139470021449292389, −1.66190035772317680079072420140, 0,
1.66190035772317680079072420140, 2.47072324790139470021449292389, 4.08905775562965259187456852663, 4.90726479932843520709803708219, 5.59077284795149942167951913297, 6.00832734024824466460998077874, 6.64494291065885444291245960204, 7.28852276793829001896923781277, 8.721918445741363563832212642809