| L(s) = 1 | − 2·7-s − 13-s − 4·17-s − 6·19-s − 6·23-s − 4·29-s − 8·31-s + 6·37-s − 6·41-s + 4·43-s − 8·47-s − 3·49-s + 2·53-s − 2·61-s − 4·67-s − 8·71-s − 16·79-s + 4·83-s + 6·89-s + 2·91-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.277·13-s − 0.970·17-s − 1.37·19-s − 1.25·23-s − 0.742·29-s − 1.43·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.256·61-s − 0.488·67-s − 0.949·71-s − 1.80·79-s + 0.439·83-s + 0.635·89-s + 0.209·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.06437636592419, −14.66298338244055, −14.14771562579914, −13.40160570959960, −12.97946781448457, −12.76540707099713, −12.04808650046730, −11.45317179012035, −11.01588753111677, −10.35344168116088, −9.993729252912920, −9.249020011394879, −8.987689565497059, −8.242359762705156, −7.754221867528209, −7.036337811807825, −6.557954301568362, −6.045398000672229, −5.515913367093574, −4.653743051902881, −4.138454399108794, −3.609722278095429, −2.792682400244377, −2.141693346995773, −1.544494075464977, 0, 0,
1.544494075464977, 2.141693346995773, 2.792682400244377, 3.609722278095429, 4.138454399108794, 4.653743051902881, 5.515913367093574, 6.045398000672229, 6.557954301568362, 7.036337811807825, 7.754221867528209, 8.242359762705156, 8.987689565497059, 9.249020011394879, 9.993729252912920, 10.35344168116088, 11.01588753111677, 11.45317179012035, 12.04808650046730, 12.76540707099713, 12.97946781448457, 13.40160570959960, 14.14771562579914, 14.66298338244055, 15.06437636592419
