| L(s) = 1 | + 1.61·2-s − 3-s + 0.618·4-s − 1.61·6-s − 7-s − 2.23·8-s + 9-s − 4.80·11-s − 0.618·12-s − 13-s − 1.61·14-s − 4.85·16-s − 3.24·17-s + 1.61·18-s − 0.891·19-s + 21-s − 7.77·22-s − 1.41·23-s + 2.23·24-s − 1.61·26-s − 27-s − 0.618·28-s + 0.459·29-s + 0.340·31-s − 3.38·32-s + 4.80·33-s − 5.25·34-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.660·6-s − 0.377·7-s − 0.790·8-s + 0.333·9-s − 1.44·11-s − 0.178·12-s − 0.277·13-s − 0.432·14-s − 1.21·16-s − 0.787·17-s + 0.381·18-s − 0.204·19-s + 0.218·21-s − 1.65·22-s − 0.294·23-s + 0.456·24-s − 0.317·26-s − 0.192·27-s − 0.116·28-s + 0.0853·29-s + 0.0611·31-s − 0.597·32-s + 0.836·33-s − 0.900·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.306380945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.306380945\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 + 0.891T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 0.459T + 29T^{2} \) |
| 31 | \( 1 - 0.340T + 31T^{2} \) |
| 37 | \( 1 - 1.71T + 37T^{2} \) |
| 41 | \( 1 + 3.07T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 4.50T + 47T^{2} \) |
| 53 | \( 1 - 0.405T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 - 7.57T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 + 9.07T + 79T^{2} \) |
| 83 | \( 1 - 3.83T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.74462777689971082455414463372, −7.10915997949955460392440260353, −6.21375563216427275777189307013, −5.81519032203386797208170314690, −5.03706162329560723232659510088, −4.54277073799381581772794460793, −3.77720038212870484144114248507, −2.82889362995771543324737867077, −2.20804211714064584970428923858, −0.47403569632589382266361033011,
0.47403569632589382266361033011, 2.20804211714064584970428923858, 2.82889362995771543324737867077, 3.77720038212870484144114248507, 4.54277073799381581772794460793, 5.03706162329560723232659510088, 5.81519032203386797208170314690, 6.21375563216427275777189307013, 7.10915997949955460392440260353, 7.74462777689971082455414463372
