| L(s) = 1 | − 2.44·2-s − 3-s + 3.98·4-s + 4.17·5-s + 2.44·6-s + 4.04·7-s − 4.86·8-s + 9-s − 10.2·10-s − 3.16·11-s − 3.98·12-s + 5.06·13-s − 9.89·14-s − 4.17·15-s + 3.92·16-s − 4.61·17-s − 2.44·18-s − 3.07·19-s + 16.6·20-s − 4.04·21-s + 7.75·22-s − 2.44·23-s + 4.86·24-s + 12.4·25-s − 12.3·26-s − 27-s + 16.1·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 0.577·3-s + 1.99·4-s + 1.86·5-s + 0.998·6-s + 1.52·7-s − 1.71·8-s + 0.333·9-s − 3.23·10-s − 0.955·11-s − 1.15·12-s + 1.40·13-s − 2.64·14-s − 1.07·15-s + 0.981·16-s − 1.12·17-s − 0.576·18-s − 0.704·19-s + 3.72·20-s − 0.882·21-s + 1.65·22-s − 0.510·23-s + 0.992·24-s + 2.48·25-s − 2.43·26-s − 0.192·27-s + 3.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.111314634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.111314634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 677 | \( 1 - T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 5 | \( 1 - 4.17T + 5T^{2} \) |
| 7 | \( 1 - 4.04T + 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 - 5.06T + 13T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 5.19T + 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 + 0.182T + 47T^{2} \) |
| 53 | \( 1 + 5.75T + 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 8.98T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 - 9.61T + 89T^{2} \) |
| 97 | \( 1 - 1.47T + 97T^{2} \) |
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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
−9.042382149253326750962898923988, −8.560213964741350986784880252041, −7.87418938633542926912291690899, −6.82030925229135851173917061525, −6.15281627180118183606151865650, −5.47156318396224271967856205696, −4.50719147171573793413334662509, −2.40980528107974115371359866099, −1.88270694612818583446340746626, −0.995554753077724557140177418366,
0.995554753077724557140177418366, 1.88270694612818583446340746626, 2.40980528107974115371359866099, 4.50719147171573793413334662509, 5.47156318396224271967856205696, 6.15281627180118183606151865650, 6.82030925229135851173917061525, 7.87418938633542926912291690899, 8.560213964741350986784880252041, 9.042382149253326750962898923988
