| L(s) = 1 | + 0.347·2-s − 0.879·4-s − 1.37·5-s − 0.652·8-s + 9-s − 0.476·10-s + 1.94·11-s − 1.98·13-s + 0.652·16-s − 0.573·17-s + 0.347·18-s + 1.20·20-s + 0.675·22-s + 1.78·23-s + 0.883·25-s − 0.689·26-s + 1.53·31-s + 0.879·32-s − 0.199·34-s − 0.879·36-s + 0.895·40-s + 0.792·41-s + 0.792·43-s − 1.71·44-s − 1.37·45-s + 0.620·46-s + 1.19·47-s + ⋯ |
| L(s) = 1 | + 0.347·2-s − 0.879·4-s − 1.37·5-s − 0.652·8-s + 9-s − 0.476·10-s + 1.94·11-s − 1.98·13-s + 0.652·16-s − 0.573·17-s + 0.347·18-s + 1.20·20-s + 0.675·22-s + 1.78·23-s + 0.883·25-s − 0.689·26-s + 1.53·31-s + 0.879·32-s − 0.199·34-s − 0.879·36-s + 0.895·40-s + 0.792·41-s + 0.792·43-s − 1.71·44-s − 1.37·45-s + 0.620·46-s + 1.19·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9114260251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9114260251\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1759 | \( 1 - T \) |
| good | 2 | \( 1 - 0.347T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.37T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.94T + T^{2} \) |
| 13 | \( 1 + 1.98T + T^{2} \) |
| 17 | \( 1 + 0.573T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.78T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.53T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.792T + T^{2} \) |
| 43 | \( 1 - 0.792T + T^{2} \) |
| 47 | \( 1 - 1.19T + T^{2} \) |
| 53 | \( 1 + 1.67T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 0.116T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.295971521982008766814587572220, −8.920778714052252183884633849786, −7.77732328961424631161915605219, −7.17610508630550953224151487549, −6.46792652612837103932398742148, −5.01251784524212718705617177703, −4.35843708139121537068077298441, −4.00838018640125575917342800611, −2.83808812683631469716646747113, −0.968888773396792202436930200982,
0.968888773396792202436930200982, 2.83808812683631469716646747113, 4.00838018640125575917342800611, 4.35843708139121537068077298441, 5.01251784524212718705617177703, 6.46792652612837103932398742148, 7.17610508630550953224151487549, 7.77732328961424631161915605219, 8.920778714052252183884633849786, 9.295971521982008766814587572220
