L(s) = 1 | − 1.48·2-s + 2.99·3-s + 0.218·4-s + 0.692·5-s − 4.46·6-s − 1.15·7-s + 2.65·8-s + 5.98·9-s − 1.03·10-s − 2.69·11-s + 0.655·12-s + 1.71·14-s + 2.07·15-s − 4.38·16-s + 4.04·17-s − 8.91·18-s − 5.06·19-s + 0.151·20-s − 3.45·21-s + 4.01·22-s + 1.01·23-s + 7.95·24-s − 4.52·25-s + 8.94·27-s − 0.251·28-s − 8.56·29-s − 3.09·30-s + ⋯ |
L(s) = 1 | − 1.05·2-s + 1.73·3-s + 0.109·4-s + 0.309·5-s − 1.82·6-s − 0.435·7-s + 0.938·8-s + 1.99·9-s − 0.326·10-s − 0.812·11-s + 0.189·12-s + 0.458·14-s + 0.536·15-s − 1.09·16-s + 0.980·17-s − 2.10·18-s − 1.16·19-s + 0.0338·20-s − 0.753·21-s + 0.855·22-s + 0.212·23-s + 1.62·24-s − 0.904·25-s + 1.72·27-s − 0.0476·28-s − 1.59·29-s − 0.564·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 3 | \( 1 - 2.99T + 3T^{2} \) |
| 5 | \( 1 - 0.692T + 5T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 17 | \( 1 - 4.04T + 17T^{2} \) |
| 19 | \( 1 + 5.06T + 19T^{2} \) |
| 23 | \( 1 - 1.01T + 23T^{2} \) |
| 29 | \( 1 + 8.56T + 29T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 3.63T + 43T^{2} \) |
| 47 | \( 1 - 0.848T + 47T^{2} \) |
| 53 | \( 1 - 1.67T + 53T^{2} \) |
| 59 | \( 1 + 1.68T + 59T^{2} \) |
| 61 | \( 1 - 2.00T + 61T^{2} \) |
| 67 | \( 1 + 8.32T + 67T^{2} \) |
| 71 | \( 1 + 5.15T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 0.873T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 3.90T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.942553676142320980328860063584, −7.58141307520595233758614327116, −6.85474921053335068545030419923, −5.73955536256514967746465060114, −4.74828992929971634598922509180, −3.82144534924980896820374411075, −3.17108387623622104839942682571, −2.15327530666983735484896126536, −1.59529824775446997348684160442, 0,
1.59529824775446997348684160442, 2.15327530666983735484896126536, 3.17108387623622104839942682571, 3.82144534924980896820374411075, 4.74828992929971634598922509180, 5.73955536256514967746465060114, 6.85474921053335068545030419923, 7.58141307520595233758614327116, 7.942553676142320980328860063584