| L(s) = 1 | − 5-s − 4·7-s − 3·9-s + 4·11-s + 2·13-s − 2·17-s − 4·19-s − 4·23-s + 25-s + 2·29-s + 8·31-s + 4·35-s − 6·41-s + 8·43-s + 3·45-s + 4·47-s + 9·49-s + 6·53-s − 4·55-s + 4·59-s + 2·61-s + 12·63-s − 2·65-s + 8·67-s − 6·73-s − 16·77-s + 9·81-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.676·35-s − 0.937·41-s + 1.21·43-s + 0.447·45-s + 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s + 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.248·65-s + 0.977·67-s − 0.702·73-s − 1.82·77-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9761436294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9761436294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.35418601613661, −13.94851198829621, −13.34593645629723, −12.98341723628689, −12.15760987415144, −12.02695672838719, −11.43896442182852, −10.85545709508886, −10.25602394769939, −9.801572803328097, −9.132489265386210, −8.679298625382222, −8.400846249873926, −7.587094274101973, −6.795911479719749, −6.438820370766750, −6.127149266759433, −5.473745902828985, −4.523478272497285, −3.905285067769871, −3.628921291984336, −2.771649712540567, −2.348105864573997, −1.185831784760264, −0.3700994540354307,
0.3700994540354307, 1.185831784760264, 2.348105864573997, 2.771649712540567, 3.628921291984336, 3.905285067769871, 4.523478272497285, 5.473745902828985, 6.127149266759433, 6.438820370766750, 6.795911479719749, 7.587094274101973, 8.400846249873926, 8.679298625382222, 9.132489265386210, 9.801572803328097, 10.25602394769939, 10.85545709508886, 11.43896442182852, 12.02695672838719, 12.15760987415144, 12.98341723628689, 13.34593645629723, 13.94851198829621, 14.35418601613661
