L(s) = 1 | − 0.618·2-s − 1.61·4-s + 5-s + 1.23·7-s + 2.23·8-s − 0.618·10-s − 3.23·11-s − 13-s − 0.763·14-s + 1.85·16-s + 0.236·17-s − 5.47·19-s − 1.61·20-s + 2.00·22-s + 1.47·23-s + 25-s + 0.618·26-s − 2.00·28-s + 5.23·29-s − 3·31-s − 5.61·32-s − 0.145·34-s + 1.23·35-s − 6·37-s + 3.38·38-s + 2.23·40-s − 10.4·41-s + ⋯ |
L(s) = 1 | − 0.437·2-s − 0.809·4-s + 0.447·5-s + 0.467·7-s + 0.790·8-s − 0.195·10-s − 0.975·11-s − 0.277·13-s − 0.204·14-s + 0.463·16-s + 0.0572·17-s − 1.25·19-s − 0.361·20-s + 0.426·22-s + 0.306·23-s + 0.200·25-s + 0.121·26-s − 0.377·28-s + 0.972·29-s − 0.538·31-s − 0.993·32-s − 0.0250·34-s + 0.208·35-s − 0.986·37-s + 0.548·38-s + 0.353·40-s − 1.63·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 17 | \( 1 - 0.236T + 17T^{2} \) |
| 19 | \( 1 + 5.47T + 19T^{2} \) |
| 23 | \( 1 - 1.47T + 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 - 6.70T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 11T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 8.76T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 6.70T + 79T^{2} \) |
| 83 | \( 1 - 8.23T + 83T^{2} \) |
| 89 | \( 1 + 9.70T + 89T^{2} \) |
| 97 | \( 1 + 6.47T + 97T^{2} \) |
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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
−8.811131808105939576959172104322, −8.311393525815262462019885310877, −7.53387510199655712037428178449, −6.58740070013619464740533919935, −5.43518177573231588486243515645, −4.91008018531659278825814255806, −3.99269120246150146571910473633, −2.67922708913879507202262590878, −1.53635149057595003695857915883, 0,
1.53635149057595003695857915883, 2.67922708913879507202262590878, 3.99269120246150146571910473633, 4.91008018531659278825814255806, 5.43518177573231588486243515645, 6.58740070013619464740533919935, 7.53387510199655712037428178449, 8.311393525815262462019885310877, 8.811131808105939576959172104322