| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s − 13-s + 16-s + 17-s + 6·22-s − 3·23-s + 26-s − 7·31-s − 32-s − 34-s − 2·37-s − 5·41-s + 6·43-s − 6·44-s + 3·46-s − 47-s − 52-s − 8·59-s − 6·61-s + 7·62-s + 64-s + 2·67-s + 68-s − 71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 0.277·13-s + 1/4·16-s + 0.242·17-s + 1.27·22-s − 0.625·23-s + 0.196·26-s − 1.25·31-s − 0.176·32-s − 0.171·34-s − 0.328·37-s − 0.780·41-s + 0.914·43-s − 0.904·44-s + 0.442·46-s − 0.145·47-s − 0.138·52-s − 1.04·59-s − 0.768·61-s + 0.889·62-s + 1/8·64-s + 0.244·67-s + 0.121·68-s − 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−13.26587449827458, −12.56352893949553, −12.29561434967622, −11.85481044507721, −11.13093292166689, −10.71678473338403, −10.57412266675774, −9.945223359258044, −9.500882867085910, −9.164622088243539, −8.276859637133018, −8.250900461425988, −7.679819288676753, −7.162218611172552, −6.899378067960309, −5.972432155988902, −5.718470630717913, −5.216986932662220, −4.651841527492924, −4.052255496144858, −3.254269293601533, −2.936725875301879, −2.233983817638776, −1.838747090494472, −1.057829290944136, 0, 0,
1.057829290944136, 1.838747090494472, 2.233983817638776, 2.936725875301879, 3.254269293601533, 4.052255496144858, 4.651841527492924, 5.216986932662220, 5.718470630717913, 5.972432155988902, 6.899378067960309, 7.162218611172552, 7.679819288676753, 8.250900461425988, 8.276859637133018, 9.164622088243539, 9.500882867085910, 9.945223359258044, 10.57412266675774, 10.71678473338403, 11.13093292166689, 11.85481044507721, 12.29561434967622, 12.56352893949553, 13.26587449827458
