| L(s) = 1 | + 3.24·3-s + 5-s − 3.24·7-s + 7.55·9-s + 1.05·11-s + 13-s + 3.24·15-s + 17-s + 4·19-s − 10.5·21-s + 2.94·23-s − 4·25-s + 14.8·27-s − 7.43·29-s + 5.05·31-s + 3.43·33-s − 3.24·35-s + 6.55·37-s + 3.24·39-s + 9.43·41-s − 0.307·43-s + 7.55·45-s + 6.80·47-s + 3.55·49-s + 3.24·51-s − 1.55·53-s + 1.05·55-s + ⋯ |
| L(s) = 1 | + 1.87·3-s + 0.447·5-s − 1.22·7-s + 2.51·9-s + 0.319·11-s + 0.277·13-s + 0.838·15-s + 0.242·17-s + 0.917·19-s − 2.30·21-s + 0.613·23-s − 0.800·25-s + 2.84·27-s − 1.38·29-s + 0.908·31-s + 0.598·33-s − 0.549·35-s + 1.07·37-s + 0.520·39-s + 1.47·41-s − 0.0469·43-s + 1.12·45-s + 0.992·47-s + 0.508·49-s + 0.454·51-s − 0.213·53-s + 0.142·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.002022993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.002022993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 2.94T + 23T^{2} \) |
| 29 | \( 1 + 7.43T + 29T^{2} \) |
| 31 | \( 1 - 5.05T + 31T^{2} \) |
| 37 | \( 1 - 6.55T + 37T^{2} \) |
| 41 | \( 1 - 9.43T + 41T^{2} \) |
| 43 | \( 1 + 0.307T + 43T^{2} \) |
| 47 | \( 1 - 6.80T + 47T^{2} \) |
| 53 | \( 1 + 1.55T + 53T^{2} \) |
| 59 | \( 1 + 5.67T + 59T^{2} \) |
| 61 | \( 1 + 9.67T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 + 3.55T + 73T^{2} \) |
| 79 | \( 1 + 6.73T + 79T^{2} \) |
| 83 | \( 1 + 2.49T + 83T^{2} \) |
| 89 | \( 1 + 2.11T + 89T^{2} \) |
| 97 | \( 1 - 7.67T + 97T^{2} \) |
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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
−8.894214501206698848234529587662, −7.76341472912627878750259728197, −7.46808960435459178987553422135, −6.49002384512892117086908621643, −5.77325324823499474225605542487, −4.44168687288280321682888152810, −3.64426033802773433597488034289, −3.04596967594206666607446461009, −2.31069467083911540355938997155, −1.18543083165333676542033689615,
1.18543083165333676542033689615, 2.31069467083911540355938997155, 3.04596967594206666607446461009, 3.64426033802773433597488034289, 4.44168687288280321682888152810, 5.77325324823499474225605542487, 6.49002384512892117086908621643, 7.46808960435459178987553422135, 7.76341472912627878750259728197, 8.894214501206698848234529587662
