| L(s) = 1 | − 2·4-s + 5-s − 7-s + 11-s + 4·13-s + 4·16-s − 2·17-s − 2·20-s − 2·23-s − 4·25-s + 2·28-s − 4·29-s − 31-s − 35-s − 2·37-s − 9·41-s + 6·43-s − 2·44-s − 47-s + 49-s − 8·52-s + 53-s + 55-s − 4·59-s + 5·61-s − 8·64-s + 4·65-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.377·7-s + 0.301·11-s + 1.10·13-s + 16-s − 0.485·17-s − 0.447·20-s − 0.417·23-s − 4/5·25-s + 0.377·28-s − 0.742·29-s − 0.179·31-s − 0.169·35-s − 0.328·37-s − 1.40·41-s + 0.914·43-s − 0.301·44-s − 0.145·47-s + 1/7·49-s − 1.10·52-s + 0.137·53-s + 0.134·55-s − 0.520·59-s + 0.640·61-s − 64-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9955433521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9955433521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−13.02912175488107, −12.49305865654027, −11.99241136353406, −11.43008532887889, −11.04652551150454, −10.25558157528314, −10.18729946304085, −9.530911554735657, −9.037203516100628, −8.855672802237086, −8.230835442919810, −7.822539609288797, −7.152260935302922, −6.636935389265365, −5.984170677562061, −5.791500806807265, −5.229462680458413, −4.528301103808082, −4.120144716899181, −3.553068457653658, −3.239893717402277, −2.311495666334532, −1.708449735801261, −1.148350169418104, −0.2935355555569731,
0.2935355555569731, 1.148350169418104, 1.708449735801261, 2.311495666334532, 3.239893717402277, 3.553068457653658, 4.120144716899181, 4.528301103808082, 5.229462680458413, 5.791500806807265, 5.984170677562061, 6.636935389265365, 7.152260935302922, 7.822539609288797, 8.230835442919810, 8.855672802237086, 9.037203516100628, 9.530911554735657, 10.18729946304085, 10.25558157528314, 11.04652551150454, 11.43008532887889, 11.99241136353406, 12.49305865654027, 13.02912175488107
