| L(s) = 1 | + 2-s − 2·3-s + 4-s + 1.61·5-s − 2·6-s − 4.85·7-s + 8-s + 9-s + 1.61·10-s − 2·12-s + 2.47·13-s − 4.85·14-s − 3.23·15-s + 16-s + 3.38·17-s + 18-s + 19-s + 1.61·20-s + 9.70·21-s − 7.85·23-s − 2·24-s − 2.38·25-s + 2.47·26-s + 4·27-s − 4.85·28-s + 10.4·29-s − 3.23·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 0.5·4-s + 0.723·5-s − 0.816·6-s − 1.83·7-s + 0.353·8-s + 0.333·9-s + 0.511·10-s − 0.577·12-s + 0.685·13-s − 1.29·14-s − 0.835·15-s + 0.250·16-s + 0.820·17-s + 0.235·18-s + 0.229·19-s + 0.361·20-s + 2.11·21-s − 1.63·23-s − 0.408·24-s − 0.476·25-s + 0.484·26-s + 0.769·27-s − 0.917·28-s + 1.94·29-s − 0.590·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 4.85T + 7T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 17 | \( 1 - 3.38T + 17T^{2} \) |
| 23 | \( 1 + 7.85T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 - 1.38T + 47T^{2} \) |
| 53 | \( 1 + 4.76T + 53T^{2} \) |
| 59 | \( 1 + 3.52T + 59T^{2} \) |
| 61 | \( 1 + 9.56T + 61T^{2} \) |
| 67 | \( 1 - 2.76T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 9.23T + 97T^{2} \) |
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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
−7.78170163991278295346999971229, −6.61837419657795573612745364479, −6.34264032383615264793978943191, −5.91331107784492094247068093318, −5.24876326650737583986214761280, −4.25174262966563049299176686504, −3.35575911344486109989536598598, −2.68148181105250630854910565410, −1.30808828819426220615574303115, 0,
1.30808828819426220615574303115, 2.68148181105250630854910565410, 3.35575911344486109989536598598, 4.25174262966563049299176686504, 5.24876326650737583986214761280, 5.91331107784492094247068093318, 6.34264032383615264793978943191, 6.61837419657795573612745364479, 7.78170163991278295346999971229
