| L(s) = 1 | − 4·7-s − 4·11-s − 13-s − 8·17-s + 8·19-s − 8·23-s + 4·29-s − 31-s − 3·37-s − 11·43-s − 8·47-s + 9·49-s + 12·53-s − 8·59-s − 2·61-s + 11·67-s − 12·71-s − 9·73-s + 16·77-s − 9·79-s − 4·83-s − 12·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s − 0.277·13-s − 1.94·17-s + 1.83·19-s − 1.66·23-s + 0.742·29-s − 0.179·31-s − 0.493·37-s − 1.67·43-s − 1.16·47-s + 9/7·49-s + 1.64·53-s − 1.04·59-s − 0.256·61-s + 1.34·67-s − 1.42·71-s − 1.05·73-s + 1.82·77-s − 1.01·79-s − 0.439·83-s − 1.27·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4374510501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4374510501\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.19592239925091, −16.01981955473450, −15.59807427249911, −15.02650764618436, −13.95232235425995, −13.66530800451871, −13.09713992189322, −12.66531977557077, −11.84931058509176, −11.47502802522811, −10.49718422843479, −10.01369576129577, −9.703663240770388, −8.847544559065369, −8.302015086141494, −7.446635328713553, −6.930041465423665, −6.287251950110847, −5.624666868826840, −4.904735174304450, −4.108063538838288, −3.207442206302595, −2.738894017229128, −1.821508352538085, −0.2872664843499359,
0.2872664843499359, 1.821508352538085, 2.738894017229128, 3.207442206302595, 4.108063538838288, 4.904735174304450, 5.624666868826840, 6.287251950110847, 6.930041465423665, 7.446635328713553, 8.302015086141494, 8.847544559065369, 9.703663240770388, 10.01369576129577, 10.49718422843479, 11.47502802522811, 11.84931058509176, 12.66531977557077, 13.09713992189322, 13.66530800451871, 13.95232235425995, 15.02650764618436, 15.59807427249911, 16.01981955473450, 16.19592239925091
