| L(s) = 1 | − 4.04·5-s − 7-s + 3.98·11-s + 6.74·13-s + 4.92·17-s − 1.24·19-s + 23-s + 11.3·25-s − 6.48·29-s − 8.38·31-s + 4.04·35-s − 7.61·37-s + 7.11·41-s − 1.11·43-s − 8.31·47-s + 49-s + 4.77·53-s − 16.1·55-s − 1.31·59-s + 0.838·61-s − 27.2·65-s + 12.4·67-s − 10.0·71-s + 6.46·73-s − 3.98·77-s − 6.45·79-s + 5.59·83-s + ⋯ |
| L(s) = 1 | − 1.80·5-s − 0.377·7-s + 1.20·11-s + 1.87·13-s + 1.19·17-s − 0.286·19-s + 0.208·23-s + 2.26·25-s − 1.20·29-s − 1.50·31-s + 0.683·35-s − 1.25·37-s + 1.11·41-s − 0.170·43-s − 1.21·47-s + 0.142·49-s + 0.655·53-s − 2.17·55-s − 0.171·59-s + 0.107·61-s − 3.38·65-s + 1.51·67-s − 1.19·71-s + 0.757·73-s − 0.454·77-s − 0.726·79-s + 0.613·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.408446134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.408446134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 4.04T + 5T^{2} \) |
| 11 | \( 1 - 3.98T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 - 4.92T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 + 8.38T + 31T^{2} \) |
| 37 | \( 1 + 7.61T + 37T^{2} \) |
| 41 | \( 1 - 7.11T + 41T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 + 8.31T + 47T^{2} \) |
| 53 | \( 1 - 4.77T + 53T^{2} \) |
| 59 | \( 1 + 1.31T + 59T^{2} \) |
| 61 | \( 1 - 0.838T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 6.46T + 73T^{2} \) |
| 79 | \( 1 + 6.45T + 79T^{2} \) |
| 83 | \( 1 - 5.59T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.035402853870743090930598825660, −7.48198243343493769707615643819, −6.77902145816991072535735422894, −6.08432158393064245986255517697, −5.21394393445992389795509231907, −4.08742020761970913605406600385, −3.62318392812414293343339285882, −3.35547522016261016989716865125, −1.62236499770347827639997107399, −0.66471793267666138791227303157,
0.66471793267666138791227303157, 1.62236499770347827639997107399, 3.35547522016261016989716865125, 3.62318392812414293343339285882, 4.08742020761970913605406600385, 5.21394393445992389795509231907, 6.08432158393064245986255517697, 6.77902145816991072535735422894, 7.48198243343493769707615643819, 8.035402853870743090930598825660
