| L(s) = 1 | + 3.34·3-s − 1.55·5-s − 5.16·7-s + 8.18·9-s + 11-s − 5.65·13-s − 5.19·15-s + 1.58·17-s + 0.0288·19-s − 17.2·21-s + 7.85·23-s − 2.58·25-s + 17.3·27-s + 2.65·29-s + 8.68·31-s + 3.34·33-s + 8.01·35-s + 9.07·37-s − 18.9·39-s − 1.21·41-s − 4.91·43-s − 12.7·45-s − 6.66·47-s + 19.6·49-s + 5.29·51-s + 0.215·53-s − 1.55·55-s + ⋯ |
| L(s) = 1 | + 1.93·3-s − 0.694·5-s − 1.95·7-s + 2.72·9-s + 0.301·11-s − 1.56·13-s − 1.34·15-s + 0.383·17-s + 0.00661·19-s − 3.76·21-s + 1.63·23-s − 0.517·25-s + 3.34·27-s + 0.493·29-s + 1.56·31-s + 0.582·33-s + 1.35·35-s + 1.49·37-s − 3.02·39-s − 0.190·41-s − 0.750·43-s − 1.89·45-s − 0.972·47-s + 2.80·49-s + 0.741·51-s + 0.0296·53-s − 0.209·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.830133230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.830133230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 - 3.34T + 3T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 7 | \( 1 + 5.16T + 7T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 0.0288T + 19T^{2} \) |
| 23 | \( 1 - 7.85T + 23T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 - 8.68T + 31T^{2} \) |
| 37 | \( 1 - 9.07T + 37T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 + 4.91T + 43T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 - 0.215T + 53T^{2} \) |
| 59 | \( 1 - 0.410T + 59T^{2} \) |
| 61 | \( 1 - 0.112T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 7.13T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.081547375277739163152688742964, −7.40919065549077240904089000717, −6.94766868463786280261024483623, −6.30117563976645047993948284932, −4.85153171426342082747052179257, −4.18800772638895789290758885398, −3.26825094771044247522563297591, −3.03165234385944966702207347555, −2.28813191912698815477978068586, −0.78352133811077426451576301949,
0.78352133811077426451576301949, 2.28813191912698815477978068586, 3.03165234385944966702207347555, 3.26825094771044247522563297591, 4.18800772638895789290758885398, 4.85153171426342082747052179257, 6.30117563976645047993948284932, 6.94766868463786280261024483623, 7.40919065549077240904089000717, 8.081547375277739163152688742964
