| L(s) = 1 | − 0.579·2-s − 1.89·3-s − 1.66·4-s + 1.47·5-s + 1.09·6-s + 2.12·8-s + 0.579·9-s − 0.854·10-s + 0.579·11-s + 3.14·12-s − 2.78·15-s + 2.09·16-s − 1.19·17-s − 0.336·18-s − 0.460·19-s − 2.45·20-s − 0.336·22-s + 2.36·23-s − 4.01·24-s − 2.82·25-s + 4.57·27-s − 6.89·29-s + 1.61·30-s + 4.44·31-s − 5.46·32-s − 1.09·33-s + 0.693·34-s + ⋯ |
| L(s) = 1 | − 0.409·2-s − 1.09·3-s − 0.831·4-s + 0.659·5-s + 0.447·6-s + 0.751·8-s + 0.193·9-s − 0.270·10-s + 0.174·11-s + 0.908·12-s − 0.719·15-s + 0.523·16-s − 0.290·17-s − 0.0792·18-s − 0.105·19-s − 0.548·20-s − 0.0716·22-s + 0.493·23-s − 0.820·24-s − 0.565·25-s + 0.881·27-s − 1.27·29-s + 0.295·30-s + 0.798·31-s − 0.965·32-s − 0.190·33-s + 0.118·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.579T + 2T^{2} \) |
| 3 | \( 1 + 1.89T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 - 0.579T + 11T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 + 0.460T + 19T^{2} \) |
| 23 | \( 1 - 2.36T + 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 9.16T + 37T^{2} \) |
| 41 | \( 1 + 4.01T + 41T^{2} \) |
| 43 | \( 1 - 8.05T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 9.39T + 53T^{2} \) |
| 59 | \( 1 - 0.240T + 59T^{2} \) |
| 61 | \( 1 - 7.72T + 61T^{2} \) |
| 67 | \( 1 + 1.44T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 3.69T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2.49T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−7.54793549499548127207293395767, −6.57171955670597389932572606816, −6.04297012758547900532931507317, −5.42083067494948379046308300130, −4.78283773392039137244283983020, −4.14683537385585712216191714811, −3.09905889221511071623218983442, −1.92849402637800627637645726706, −0.977387256770598168798957325041, 0,
0.977387256770598168798957325041, 1.92849402637800627637645726706, 3.09905889221511071623218983442, 4.14683537385585712216191714811, 4.78283773392039137244283983020, 5.42083067494948379046308300130, 6.04297012758547900532931507317, 6.57171955670597389932572606816, 7.54793549499548127207293395767
