| L(s) = 1 | + 0.770·2-s − 1.40·4-s + 3.98·7-s − 2.62·8-s + 6.35·11-s + 5.35·13-s + 3.07·14-s + 0.793·16-s + 3.45·17-s + 2.32·19-s + 4.89·22-s − 0.955·23-s + 4.12·26-s − 5.61·28-s − 7.26·29-s + 6.40·31-s + 5.85·32-s + 2.66·34-s − 2.83·37-s + 1.78·38-s + 5.35·41-s + 3.93·43-s − 8.93·44-s − 0.735·46-s − 2.48·47-s + 8.90·49-s − 7.53·52-s + ⋯ |
| L(s) = 1 | + 0.544·2-s − 0.703·4-s + 1.50·7-s − 0.927·8-s + 1.91·11-s + 1.48·13-s + 0.820·14-s + 0.198·16-s + 0.838·17-s + 0.532·19-s + 1.04·22-s − 0.199·23-s + 0.808·26-s − 1.06·28-s − 1.34·29-s + 1.15·31-s + 1.03·32-s + 0.456·34-s − 0.465·37-s + 0.289·38-s + 0.835·41-s + 0.600·43-s − 1.34·44-s − 0.108·46-s − 0.362·47-s + 1.27·49-s − 1.04·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.400804845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.400804845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.770T + 2T^{2} \) |
| 7 | \( 1 - 3.98T + 7T^{2} \) |
| 11 | \( 1 - 6.35T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 3.45T + 17T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 + 0.955T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 - 6.40T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 - 5.35T + 41T^{2} \) |
| 43 | \( 1 - 3.93T + 43T^{2} \) |
| 47 | \( 1 + 2.48T + 47T^{2} \) |
| 53 | \( 1 + 3.95T + 53T^{2} \) |
| 59 | \( 1 + 0.0941T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 5.81T + 71T^{2} \) |
| 73 | \( 1 - 3.79T + 73T^{2} \) |
| 79 | \( 1 + 5.44T + 79T^{2} \) |
| 83 | \( 1 + 9.24T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 7.89T + 97T^{2} \) |
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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
−8.185432645204865507804198564184, −7.56629004946208303292344211132, −6.47544827882195494642407769857, −5.86598619086736680590011747605, −5.23128638832089483722044968975, −4.27696070835220236074916497320, −3.96819486651482049635237701172, −3.13055652828959637478284457173, −1.56891678457349402606069862422, −1.08313301027434652929685114872,
1.08313301027434652929685114872, 1.56891678457349402606069862422, 3.13055652828959637478284457173, 3.96819486651482049635237701172, 4.27696070835220236074916497320, 5.23128638832089483722044968975, 5.86598619086736680590011747605, 6.47544827882195494642407769857, 7.56629004946208303292344211132, 8.185432645204865507804198564184
