L(s) = 1 | − 2.12·3-s + 4.12·7-s + 1.51·9-s + 2.64·11-s − 2.51·13-s + 0.515·17-s − 19-s − 8.76·21-s + 3.09·23-s + 3.15·27-s − 7.79·29-s − 3.67·31-s − 5.60·33-s − 10.2·37-s + 5.34·39-s + 8.88·41-s + 8.64·43-s − 4.96·47-s + 10.0·49-s − 1.09·51-s − 5.48·53-s + 2.12·57-s − 3.15·59-s − 12.6·61-s + 6.24·63-s − 7.40·67-s − 6.57·69-s + ⋯ |
L(s) = 1 | − 1.22·3-s + 1.55·7-s + 0.505·9-s + 0.795·11-s − 0.697·13-s + 0.124·17-s − 0.229·19-s − 1.91·21-s + 0.645·23-s + 0.607·27-s − 1.44·29-s − 0.659·31-s − 0.976·33-s − 1.68·37-s + 0.855·39-s + 1.38·41-s + 1.31·43-s − 0.724·47-s + 1.43·49-s − 0.153·51-s − 0.753·53-s + 0.281·57-s − 0.410·59-s − 1.61·61-s + 0.787·63-s − 0.904·67-s − 0.791·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2.12T + 3T^{2} \) |
| 7 | \( 1 - 4.12T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 2.51T + 13T^{2} \) |
| 17 | \( 1 - 0.515T + 17T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + 3.67T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 + 4.96T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 + 3.15T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 7.40T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 3.28T + 83T^{2} \) |
| 89 | \( 1 + 7.60T + 89T^{2} \) |
| 97 | \( 1 + 3.93T + 97T^{2} \) |
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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
−7.34822679196033284694041635216, −6.95216535452745414857162855637, −5.78715900868607695906073857784, −5.62815195642420023889981657270, −4.64291553500806712231999541686, −4.38049790995996931726150682332, −3.16107373356181429298660507306, −1.92037791321894298235650752660, −1.26257788990635774384837358460, 0,
1.26257788990635774384837358460, 1.92037791321894298235650752660, 3.16107373356181429298660507306, 4.38049790995996931726150682332, 4.64291553500806712231999541686, 5.62815195642420023889981657270, 5.78715900868607695906073857784, 6.95216535452745414857162855637, 7.34822679196033284694041635216