L(s) = 1 | + 2-s + 2.58·3-s + 4-s − 3.96·5-s + 2.58·6-s − 1.63·7-s + 8-s + 3.69·9-s − 3.96·10-s + 0.219·11-s + 2.58·12-s + 4.10·13-s − 1.63·14-s − 10.2·15-s + 16-s − 5.27·17-s + 3.69·18-s − 6.25·19-s − 3.96·20-s − 4.23·21-s + 0.219·22-s − 5.57·23-s + 2.58·24-s + 10.7·25-s + 4.10·26-s + 1.78·27-s − 1.63·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.49·3-s + 0.5·4-s − 1.77·5-s + 1.05·6-s − 0.618·7-s + 0.353·8-s + 1.23·9-s − 1.25·10-s + 0.0660·11-s + 0.746·12-s + 1.13·13-s − 0.437·14-s − 2.64·15-s + 0.250·16-s − 1.27·17-s + 0.870·18-s − 1.43·19-s − 0.887·20-s − 0.923·21-s + 0.0467·22-s − 1.16·23-s + 0.528·24-s + 2.14·25-s + 0.805·26-s + 0.344·27-s − 0.309·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 2.58T + 3T^{2} \) |
| 5 | \( 1 + 3.96T + 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 - 0.219T + 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 + 5.27T + 17T^{2} \) |
| 19 | \( 1 + 6.25T + 19T^{2} \) |
| 23 | \( 1 + 5.57T + 23T^{2} \) |
| 29 | \( 1 + 0.0521T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 - 8.03T + 41T^{2} \) |
| 43 | \( 1 + 9.17T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 0.599T + 53T^{2} \) |
| 59 | \( 1 + 5.07T + 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 67 | \( 1 + 6.77T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 6.99T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 5.63T + 89T^{2} \) |
| 97 | \( 1 - 3.53T + 97T^{2} \) |
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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
−8.214714492552152646443772679441, −7.38325758634914151444207049229, −6.77993572913060383667543549414, −5.99384238825533588907346559133, −4.53057059853666100057231560075, −3.98726747282669451772315593074, −3.61075858225654510718832486516, −2.80432670351924747396065389983, −1.83929892157277117299193985347, 0,
1.83929892157277117299193985347, 2.80432670351924747396065389983, 3.61075858225654510718832486516, 3.98726747282669451772315593074, 4.53057059853666100057231560075, 5.99384238825533588907346559133, 6.77993572913060383667543549414, 7.38325758634914151444207049229, 8.214714492552152646443772679441