L(s) = 1 | − 2-s + 3-s + 4-s + 4.12·5-s − 6-s + 7-s − 8-s + 9-s − 4.12·10-s − 0.209·11-s + 12-s + 2.02·13-s − 14-s + 4.12·15-s + 16-s − 6.22·17-s − 18-s + 7.05·19-s + 4.12·20-s + 21-s + 0.209·22-s + 4.78·23-s − 24-s + 12.0·25-s − 2.02·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.84·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.30·10-s − 0.0631·11-s + 0.288·12-s + 0.562·13-s − 0.267·14-s + 1.06·15-s + 0.250·16-s − 1.51·17-s − 0.235·18-s + 1.61·19-s + 0.923·20-s + 0.218·21-s + 0.0446·22-s + 0.997·23-s − 0.204·24-s + 2.40·25-s − 0.397·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.212408518\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.212408518\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
good | 5 | \( 1 - 4.12T + 5T^{2} \) |
| 11 | \( 1 + 0.209T + 11T^{2} \) |
| 13 | \( 1 - 2.02T + 13T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 19 | \( 1 - 7.05T + 19T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 + 6.72T + 29T^{2} \) |
| 31 | \( 1 - 3.59T + 31T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 - 3.48T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 2.83T + 47T^{2} \) |
| 53 | \( 1 - 0.180T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 5.32T + 61T^{2} \) |
| 67 | \( 1 + 2.99T + 67T^{2} \) |
| 71 | \( 1 + 5.25T + 71T^{2} \) |
| 73 | \( 1 + 1.40T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 2.56T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.88763801398963217263220880097, −7.15381774618137305441961037895, −6.58593415168935624794610927570, −5.77739466289550756963152971871, −5.27360503651503644386051445682, −4.32100341154554580165518222859, −3.12000080647599501635546157802, −2.47447202450941407588078984856, −1.73875114708461622324652208907, −1.03826005146978867216371929103,
1.03826005146978867216371929103, 1.73875114708461622324652208907, 2.47447202450941407588078984856, 3.12000080647599501635546157802, 4.32100341154554580165518222859, 5.27360503651503644386051445682, 5.77739466289550756963152971871, 6.58593415168935624794610927570, 7.15381774618137305441961037895, 7.88763801398963217263220880097