| L(s) = 1 | − 2.35·2-s + 3.55·4-s + 5-s + 1.35·7-s − 3.66·8-s − 2.35·10-s − 1.49·11-s − 3.19·14-s + 1.52·16-s − 4.71·17-s + 5.54·19-s + 3.55·20-s + 3.52·22-s + 6.78·23-s + 25-s + 4.82·28-s − 8.04·29-s − 7.27·31-s + 3.72·32-s + 11.1·34-s + 1.35·35-s + 2.84·37-s − 13.0·38-s − 3.66·40-s − 1.63·41-s − 4.14·43-s − 5.31·44-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.77·4-s + 0.447·5-s + 0.512·7-s − 1.29·8-s − 0.745·10-s − 0.450·11-s − 0.854·14-s + 0.381·16-s − 1.14·17-s + 1.27·19-s + 0.794·20-s + 0.750·22-s + 1.41·23-s + 0.200·25-s + 0.911·28-s − 1.49·29-s − 1.30·31-s + 0.659·32-s + 1.90·34-s + 0.229·35-s + 0.467·37-s − 2.11·38-s − 0.579·40-s − 0.254·41-s − 0.632·43-s − 0.800·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + 1.49T + 11T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 - 6.78T + 23T^{2} \) |
| 29 | \( 1 + 8.04T + 29T^{2} \) |
| 31 | \( 1 + 7.27T + 31T^{2} \) |
| 37 | \( 1 - 2.84T + 37T^{2} \) |
| 41 | \( 1 + 1.63T + 41T^{2} \) |
| 43 | \( 1 + 4.14T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 - 5.39T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 7.36T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 2.89T + 83T^{2} \) |
| 89 | \( 1 - 3.93T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
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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
−7.74590268418876961808567303532, −6.99435425906805003379328774415, −6.58072238041545806235753578986, −5.39931444298427051374072285010, −5.00875372486309072726558444320, −3.72862861275838377899154887231, −2.69653358862132169579040377132, −1.93271914388934861996803277340, −1.18610395070062165827795907169, 0,
1.18610395070062165827795907169, 1.93271914388934861996803277340, 2.69653358862132169579040377132, 3.72862861275838377899154887231, 5.00875372486309072726558444320, 5.39931444298427051374072285010, 6.58072238041545806235753578986, 6.99435425906805003379328774415, 7.74590268418876961808567303532
