L(s) = 1 | + 2-s + 2.56·3-s + 4-s + 2.56·6-s + 2.56·7-s + 8-s + 3.56·9-s + 4·11-s + 2.56·12-s − 5.68·13-s + 2.56·14-s + 16-s − 3.43·17-s + 3.56·18-s − 19-s + 6.56·21-s + 4·22-s − 7.68·23-s + 2.56·24-s − 5.68·26-s + 1.43·27-s + 2.56·28-s − 5.68·29-s − 5.12·31-s + 32-s + 10.2·33-s − 3.43·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.47·3-s + 0.5·4-s + 1.04·6-s + 0.968·7-s + 0.353·8-s + 1.18·9-s + 1.20·11-s + 0.739·12-s − 1.57·13-s + 0.684·14-s + 0.250·16-s − 0.833·17-s + 0.839·18-s − 0.229·19-s + 1.43·21-s + 0.852·22-s − 1.60·23-s + 0.522·24-s − 1.11·26-s + 0.276·27-s + 0.484·28-s − 1.05·29-s − 0.920·31-s + 0.176·32-s + 1.78·33-s − 0.589·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.126092246\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.126092246\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 - 4.56T + 53T^{2} \) |
| 59 | \( 1 - 2.56T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 2.56T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 1.68T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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
−9.730747983450564972363666958319, −9.269461802051136599333512278135, −8.218481723170876935839658217475, −7.63377684795684572540054953690, −6.80217316125886040642651172131, −5.56507900355847824074066298697, −4.34525009015419484453738364839, −3.90135760216719732222356982045, −2.48187998105717011530260448552, −1.88554258137734670923623349210,
1.88554258137734670923623349210, 2.48187998105717011530260448552, 3.90135760216719732222356982045, 4.34525009015419484453738364839, 5.56507900355847824074066298697, 6.80217316125886040642651172131, 7.63377684795684572540054953690, 8.218481723170876935839658217475, 9.269461802051136599333512278135, 9.730747983450564972363666958319