L(s) = 1 | − 1.61·2-s − 3-s + 0.618·4-s + 1.61·6-s + 2.23·8-s + 9-s − 2.23·11-s − 0.618·12-s + 6.47·13-s − 4.85·16-s − 2.47·17-s − 1.61·18-s + 2.47·19-s + 3.61·22-s − 4.23·23-s − 2.23·24-s − 10.4·26-s − 27-s − 3·29-s − 4·31-s + 3.38·32-s + 2.23·33-s + 4.00·34-s + 0.618·36-s + 3.47·37-s − 4.00·38-s − 6.47·39-s + ⋯ |
L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.309·4-s + 0.660·6-s + 0.790·8-s + 0.333·9-s − 0.674·11-s − 0.178·12-s + 1.79·13-s − 1.21·16-s − 0.599·17-s − 0.381·18-s + 0.567·19-s + 0.771·22-s − 0.883·23-s − 0.456·24-s − 2.05·26-s − 0.192·27-s − 0.557·29-s − 0.718·31-s + 0.597·32-s + 0.389·33-s + 0.685·34-s + 0.103·36-s + 0.570·37-s − 0.648·38-s − 1.03·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 + 4.23T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.244112212624253705309480169298, −7.62212699278570071919436946292, −6.82155388824115498873885779416, −6.01067829036703858573026269824, −5.28535548361277769017829210721, −4.31208761114218240125196099571, −3.51068305168562228887630606089, −2.06750539763685971804813932702, −1.15578551851287966772301927106, 0,
1.15578551851287966772301927106, 2.06750539763685971804813932702, 3.51068305168562228887630606089, 4.31208761114218240125196099571, 5.28535548361277769017829210721, 6.01067829036703858573026269824, 6.82155388824115498873885779416, 7.62212699278570071919436946292, 8.244112212624253705309480169298