L(s) = 1 | − 0.414·3-s + 5-s − 2.82·9-s + 3.82·11-s − 3.58·13-s − 0.414·15-s − 6.41·17-s + 3.65·19-s − 0.585·23-s + 25-s + 2.41·27-s + 6.65·29-s + 4.58·31-s − 1.58·33-s − 3.41·37-s + 1.48·39-s − 0.585·41-s − 11.6·43-s − 2.82·45-s − 8.89·47-s + 2.65·51-s − 3.75·53-s + 3.82·55-s − 1.51·57-s + 3.41·59-s − 5.17·61-s − 3.58·65-s + ⋯ |
L(s) = 1 | − 0.239·3-s + 0.447·5-s − 0.942·9-s + 1.15·11-s − 0.994·13-s − 0.106·15-s − 1.55·17-s + 0.838·19-s − 0.122·23-s + 0.200·25-s + 0.464·27-s + 1.23·29-s + 0.823·31-s − 0.276·33-s − 0.561·37-s + 0.237·39-s − 0.0914·41-s − 1.77·43-s − 0.421·45-s − 1.29·47-s + 0.372·51-s − 0.516·53-s + 0.516·55-s − 0.200·57-s + 0.444·59-s − 0.662·61-s − 0.444·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 11 | \( 1 - 3.82T + 11T^{2} \) |
| 13 | \( 1 + 3.58T + 13T^{2} \) |
| 17 | \( 1 + 6.41T + 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 + 0.585T + 23T^{2} \) |
| 29 | \( 1 - 6.65T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 0.585T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 8.89T + 47T^{2} \) |
| 53 | \( 1 + 3.75T + 53T^{2} \) |
| 59 | \( 1 - 3.41T + 59T^{2} \) |
| 61 | \( 1 + 5.17T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 + 5.17T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.411549592356559750202384864919, −7.12027456234924130525223839857, −6.62641694810354889980586563675, −5.97784003847560697057061298479, −5.01300599103088654879026902206, −4.50247830156430512006234874105, −3.28828092338061938162100098841, −2.51549787270007322302359770170, −1.44195824761893878378737535022, 0,
1.44195824761893878378737535022, 2.51549787270007322302359770170, 3.28828092338061938162100098841, 4.50247830156430512006234874105, 5.01300599103088654879026902206, 5.97784003847560697057061298479, 6.62641694810354889980586563675, 7.12027456234924130525223839857, 8.411549592356559750202384864919