| L(s) = 1 | + 2-s + 4-s − 1.80·7-s + 8-s + 9-s + 1.24·11-s − 1.80·14-s + 16-s − 0.445·17-s + 18-s − 0.445·19-s + 1.24·22-s + 1.24·23-s + 25-s − 1.80·28-s − 0.445·29-s − 0.445·31-s + 32-s − 0.445·34-s + 36-s − 1.80·37-s − 0.445·38-s + 1.24·41-s − 1.80·43-s + 1.24·44-s + 1.24·46-s + 2.24·49-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 1.80·7-s + 8-s + 9-s + 1.24·11-s − 1.80·14-s + 16-s − 0.445·17-s + 18-s − 0.445·19-s + 1.24·22-s + 1.24·23-s + 25-s − 1.80·28-s − 0.445·29-s − 0.445·31-s + 32-s − 0.445·34-s + 36-s − 1.80·37-s − 0.445·38-s + 1.24·41-s − 1.80·43-s + 1.24·44-s + 1.24·46-s + 2.24·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.118081379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.118081379\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.80T + T^{2} \) |
| 11 | \( 1 - 1.24T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 + 0.445T + T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - 1.24T + T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 + 1.80T + T^{2} \) |
| 61 | \( 1 - 1.24T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.80T + T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.80T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.375445853321152082776943766921, −8.754895114382796941310694135034, −7.18967309368170271650745755970, −6.85796850160243507272025522402, −6.35289723487539999629854009989, −5.31893453327012316495797516672, −4.26276995996199409891652902832, −3.63280360133122603551530594142, −2.83085693221179326680520788488, −1.46625051177396352223191823842,
1.46625051177396352223191823842, 2.83085693221179326680520788488, 3.63280360133122603551530594142, 4.26276995996199409891652902832, 5.31893453327012316495797516672, 6.35289723487539999629854009989, 6.85796850160243507272025522402, 7.18967309368170271650745755970, 8.754895114382796941310694135034, 9.375445853321152082776943766921
