| L(s) = 1 | + 2-s + 1.05·3-s + 4-s − 0.665·5-s + 1.05·6-s − 2.37·7-s + 8-s − 1.88·9-s − 0.665·10-s − 4.35·11-s + 1.05·12-s + 1.81·13-s − 2.37·14-s − 0.703·15-s + 16-s + 3.85·17-s − 1.88·18-s + 7.50·19-s − 0.665·20-s − 2.50·21-s − 4.35·22-s + 6.72·23-s + 1.05·24-s − 4.55·25-s + 1.81·26-s − 5.16·27-s − 2.37·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.610·3-s + 0.5·4-s − 0.297·5-s + 0.431·6-s − 0.895·7-s + 0.353·8-s − 0.627·9-s − 0.210·10-s − 1.31·11-s + 0.305·12-s + 0.504·13-s − 0.633·14-s − 0.181·15-s + 0.250·16-s + 0.935·17-s − 0.443·18-s + 1.72·19-s − 0.148·20-s − 0.546·21-s − 0.927·22-s + 1.40·23-s + 0.215·24-s − 0.911·25-s + 0.356·26-s − 0.993·27-s − 0.447·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 - 1.05T + 3T^{2} \) |
| 5 | \( 1 + 0.665T + 5T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 - 1.81T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 - 7.50T + 19T^{2} \) |
| 23 | \( 1 - 6.72T + 23T^{2} \) |
| 29 | \( 1 + 3.95T + 29T^{2} \) |
| 31 | \( 1 - 2.60T + 31T^{2} \) |
| 37 | \( 1 + 8.15T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 4.63T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 8.25T + 59T^{2} \) |
| 61 | \( 1 + 7.31T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 + 9.05T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 6.27T + 83T^{2} \) |
| 89 | \( 1 + 2.45T + 89T^{2} \) |
| 97 | \( 1 - 2.26T + 97T^{2} \) |
| 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
−7.60870060792919978004729930333, −7.16190448763770545682654091033, −6.16944106967851758935480241924, −5.40721771265052010560744255826, −5.06499983451407229095529778445, −3.68786496625462754167897646936, −3.18790204116188872583342110598, −2.84745181124718377608520936782, −1.53424903357473829455204079113, 0,
1.53424903357473829455204079113, 2.84745181124718377608520936782, 3.18790204116188872583342110598, 3.68786496625462754167897646936, 5.06499983451407229095529778445, 5.40721771265052010560744255826, 6.16944106967851758935480241924, 7.16190448763770545682654091033, 7.60870060792919978004729930333
