L(s) = 1 | − 2.61·2-s − 0.146·3-s + 4.81·4-s + 0.383·6-s − 1.14·7-s − 7.36·8-s − 2.97·9-s − 5.36·11-s − 0.707·12-s + 2.41·13-s + 2.97·14-s + 9.58·16-s + 5.74·17-s + 7.77·18-s + 0.167·21-s + 14.0·22-s + 1.23·23-s + 1.08·24-s − 6.30·26-s + 0.877·27-s − 5.49·28-s − 3.91·29-s + 6.48·31-s − 10.3·32-s + 0.788·33-s − 15.0·34-s − 14.3·36-s + ⋯ |
L(s) = 1 | − 1.84·2-s − 0.0847·3-s + 2.40·4-s + 0.156·6-s − 0.431·7-s − 2.60·8-s − 0.992·9-s − 1.61·11-s − 0.204·12-s + 0.669·13-s + 0.796·14-s + 2.39·16-s + 1.39·17-s + 1.83·18-s + 0.0365·21-s + 2.98·22-s + 0.257·23-s + 0.220·24-s − 1.23·26-s + 0.168·27-s − 1.03·28-s − 0.726·29-s + 1.16·31-s − 1.82·32-s + 0.137·33-s − 2.57·34-s − 2.39·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 0.146T + 3T^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 + 5.36T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 + 7.59T + 43T^{2} \) |
| 47 | \( 1 + 7.85T + 47T^{2} \) |
| 53 | \( 1 + 0.0179T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 0.0189T + 61T^{2} \) |
| 67 | \( 1 - 7.07T + 67T^{2} \) |
| 71 | \( 1 + 2.73T + 71T^{2} \) |
| 73 | \( 1 + 5.58T + 73T^{2} \) |
| 79 | \( 1 + 9.56T + 79T^{2} \) |
| 83 | \( 1 - 0.390T + 83T^{2} \) |
| 89 | \( 1 + 3.57T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.83874790006435829393232272009, −6.96737299673954428015834629585, −6.07147883914154812774712392381, −5.78307063720889539785818779121, −4.79416407520752075832463403693, −3.14792555425631766415840483918, −3.01288246575488792706892284273, −1.96439191448579266652932001273, −0.893047017823122366879717555199, 0,
0.893047017823122366879717555199, 1.96439191448579266652932001273, 3.01288246575488792706892284273, 3.14792555425631766415840483918, 4.79416407520752075832463403693, 5.78307063720889539785818779121, 6.07147883914154812774712392381, 6.96737299673954428015834629585, 7.83874790006435829393232272009