L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 2·7-s + 8-s + 9-s − 10-s + 2·11-s + 12-s + 2·14-s − 15-s + 16-s − 4·17-s + 18-s + 6·19-s − 20-s + 2·21-s + 2·22-s + 6·23-s + 24-s + 25-s + 27-s + 2·28-s − 10·29-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.436·21-s + 0.426·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.85·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 218010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 218010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.23575688559112, −12.96205185109594, −12.19364077519268, −11.76913202088887, −11.52094207101863, −11.03356645355124, −10.55211990220052, −9.969476321613145, −9.296652233241083, −9.005147237200942, −8.497215869521369, −7.899175952405264, −7.478440923396102, −7.024261855609962, −6.665269905620511, −5.869826972789659, −5.406740701098029, −4.799310838836580, −4.439590286672639, −3.894930436526339, −3.347260516731929, −2.845798418746208, −2.277317377295975, −1.429957091775750, −1.188822545082491, 0,
1.188822545082491, 1.429957091775750, 2.277317377295975, 2.845798418746208, 3.347260516731929, 3.894930436526339, 4.439590286672639, 4.799310838836580, 5.406740701098029, 5.869826972789659, 6.665269905620511, 7.024261855609962, 7.478440923396102, 7.899175952405264, 8.497215869521369, 9.005147237200942, 9.296652233241083, 9.969476321613145, 10.55211990220052, 11.03356645355124, 11.52094207101863, 11.76913202088887, 12.19364077519268, 12.96205185109594, 13.23575688559112