| L(s) = 1 | − 2-s + 4-s + 2.95·5-s + 0.794·7-s − 8-s − 2.95·10-s − 3.15·11-s − 0.00112·13-s − 0.794·14-s + 16-s + 4.32·17-s + 3.12·19-s + 2.95·20-s + 3.15·22-s + 7.95·23-s + 3.72·25-s + 0.00112·26-s + 0.794·28-s + 2.72·29-s + 7.54·31-s − 32-s − 4.32·34-s + 2.34·35-s + 6.76·37-s − 3.12·38-s − 2.95·40-s + 11.8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.32·5-s + 0.300·7-s − 0.353·8-s − 0.934·10-s − 0.950·11-s − 0.000311·13-s − 0.212·14-s + 0.250·16-s + 1.04·17-s + 0.716·19-s + 0.660·20-s + 0.671·22-s + 1.65·23-s + 0.745·25-s + 0.000220·26-s + 0.150·28-s + 0.506·29-s + 1.35·31-s − 0.176·32-s − 0.741·34-s + 0.396·35-s + 1.11·37-s − 0.506·38-s − 0.467·40-s + 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.256398134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.256398134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 2.95T + 5T^{2} \) |
| 7 | \( 1 - 0.794T + 7T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 13 | \( 1 + 0.00112T + 13T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 - 7.95T + 23T^{2} \) |
| 29 | \( 1 - 2.72T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 4.75T + 43T^{2} \) |
| 47 | \( 1 - 5.13T + 47T^{2} \) |
| 53 | \( 1 + 2.65T + 53T^{2} \) |
| 59 | \( 1 + 8.78T + 59T^{2} \) |
| 61 | \( 1 + 4.01T + 61T^{2} \) |
| 67 | \( 1 + 8.23T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 5.69T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 6.40T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−7.81856718422615518172709210868, −7.32572627469474461526271386596, −6.40184680006292844336844347003, −5.81746104517372060679985917684, −5.21837731418748186451559795885, −4.49535791782021845108274370850, −2.90439168932245371490588716628, −2.79883303923852304457955512485, −1.56833168067798471055445229404, −0.895593147583580473092643648345,
0.895593147583580473092643648345, 1.56833168067798471055445229404, 2.79883303923852304457955512485, 2.90439168932245371490588716628, 4.49535791782021845108274370850, 5.21837731418748186451559795885, 5.81746104517372060679985917684, 6.40184680006292844336844347003, 7.32572627469474461526271386596, 7.81856718422615518172709210868
