| L(s) = 1 | − 2.23·2-s + 3.00·4-s + 1.41·5-s − 2.23·8-s − 3·9-s − 3.16·10-s + 3.16·13-s − 0.999·16-s − 3.16·17-s + 6.70·18-s + 6.32·19-s + 4.24·20-s − 4·23-s − 2.99·25-s − 7.07·26-s + 4.47·29-s − 8.48·31-s + 6.70·32-s + 7.07·34-s − 9.00·36-s + 8·37-s − 14.1·38-s − 3.16·40-s − 9.48·41-s + 8.94·43-s − 4.24·45-s + 8.94·46-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s + 0.632·5-s − 0.790·8-s − 9-s − 1.00·10-s + 0.877·13-s − 0.249·16-s − 0.766·17-s + 1.58·18-s + 1.45·19-s + 0.948·20-s − 0.834·23-s − 0.599·25-s − 1.38·26-s + 0.830·29-s − 1.52·31-s + 1.18·32-s + 1.21·34-s − 1.50·36-s + 1.31·37-s − 2.29·38-s − 0.500·40-s − 1.48·41-s + 1.36·43-s − 0.632·45-s + 1.31·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 9.48T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 6.32T + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−7.925673968922034493219412844279, −7.30371802714909477984665854972, −6.35804627499423289149090945378, −5.92337889431407002905744806738, −5.06115642172435863554255875009, −3.87245606050115998429099834505, −2.85787921928094201803772243166, −2.03918784027466561352795425650, −1.17670289247292846010482550586, 0,
1.17670289247292846010482550586, 2.03918784027466561352795425650, 2.85787921928094201803772243166, 3.87245606050115998429099834505, 5.06115642172435863554255875009, 5.92337889431407002905744806738, 6.35804627499423289149090945378, 7.30371802714909477984665854972, 7.925673968922034493219412844279
