| L(s) = 1 | + 2-s + 3-s + 4-s + 3.14·5-s + 6-s + 1.95·7-s + 8-s + 9-s + 3.14·10-s + 4.29·11-s + 12-s + 13-s + 1.95·14-s + 3.14·15-s + 16-s + 1.01·17-s + 18-s + 0.956·19-s + 3.14·20-s + 1.95·21-s + 4.29·22-s + 4.90·23-s + 24-s + 4.89·25-s + 26-s + 27-s + 1.95·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.40·5-s + 0.408·6-s + 0.739·7-s + 0.353·8-s + 0.333·9-s + 0.994·10-s + 1.29·11-s + 0.288·12-s + 0.277·13-s + 0.522·14-s + 0.812·15-s + 0.250·16-s + 0.245·17-s + 0.235·18-s + 0.219·19-s + 0.703·20-s + 0.426·21-s + 0.915·22-s + 1.02·23-s + 0.204·24-s + 0.979·25-s + 0.196·26-s + 0.192·27-s + 0.369·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.028740552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.028740552\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 3.14T + 5T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 19 | \( 1 - 0.956T + 19T^{2} \) |
| 23 | \( 1 - 4.90T + 23T^{2} \) |
| 29 | \( 1 + 7.40T + 29T^{2} \) |
| 31 | \( 1 + 0.956T + 31T^{2} \) |
| 37 | \( 1 - 0.238T + 37T^{2} \) |
| 41 | \( 1 + 7.48T + 41T^{2} \) |
| 43 | \( 1 + 6.52T + 43T^{2} \) |
| 47 | \( 1 - 6.09T + 47T^{2} \) |
| 53 | \( 1 + 2.16T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 7.32T + 61T^{2} \) |
| 67 | \( 1 + 0.0784T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 3.19T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 8.34T + 97T^{2} \) |
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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
−7.73703627968710032485149095008, −6.95007438736413215236176554762, −6.44042123871478412945819970763, −5.62244294777973751983426752025, −5.13672828588412262587528535724, −4.26835317011745503913613850151, −3.50654836666087893462183867276, −2.71300244377780879630124393056, −1.64284479308998410421338711793, −1.45979792308207253906949000103,
1.45979792308207253906949000103, 1.64284479308998410421338711793, 2.71300244377780879630124393056, 3.50654836666087893462183867276, 4.26835317011745503913613850151, 5.13672828588412262587528535724, 5.62244294777973751983426752025, 6.44042123871478412945819970763, 6.95007438736413215236176554762, 7.73703627968710032485149095008
