| L(s) = 1 | + 2.31·2-s + 3.33·4-s + 0.0587·7-s + 3.08·8-s − 3.08·11-s + 2.27·13-s + 0.135·14-s + 0.463·16-s − 5.08·17-s + 0.404·19-s − 7.13·22-s − 5.87·23-s + 5.26·26-s + 0.196·28-s + 5.59·29-s − 7.40·31-s − 5.10·32-s − 11.7·34-s + 3.67·37-s + 0.934·38-s + 1.19·41-s + 43-s − 10.3·44-s − 13.5·46-s + 4.35·47-s − 6.99·49-s + 7.60·52-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 1.66·4-s + 0.0222·7-s + 1.09·8-s − 0.931·11-s + 0.631·13-s + 0.0362·14-s + 0.115·16-s − 1.23·17-s + 0.0927·19-s − 1.52·22-s − 1.22·23-s + 1.03·26-s + 0.0370·28-s + 1.03·29-s − 1.32·31-s − 0.903·32-s − 2.01·34-s + 0.603·37-s + 0.151·38-s + 0.186·41-s + 0.152·43-s − 1.55·44-s − 2.00·46-s + 0.635·47-s − 0.999·49-s + 1.05·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 2.31T + 2T^{2} \) |
| 7 | \( 1 - 0.0587T + 7T^{2} \) |
| 11 | \( 1 + 3.08T + 11T^{2} \) |
| 13 | \( 1 - 2.27T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 - 0.404T + 19T^{2} \) |
| 23 | \( 1 + 5.87T + 23T^{2} \) |
| 29 | \( 1 - 5.59T + 29T^{2} \) |
| 31 | \( 1 + 7.40T + 31T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 - 1.19T + 41T^{2} \) |
| 47 | \( 1 - 4.35T + 47T^{2} \) |
| 53 | \( 1 - 6.62T + 53T^{2} \) |
| 59 | \( 1 + 3.81T + 59T^{2} \) |
| 61 | \( 1 + 9.62T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 6.86T + 73T^{2} \) |
| 79 | \( 1 + 8.09T + 79T^{2} \) |
| 83 | \( 1 + 8.97T + 83T^{2} \) |
| 89 | \( 1 - 2.13T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−7.11786914936833242200876562192, −6.36400121037507245629880474392, −5.87122648860558932637535318500, −5.25394383625954075823188557944, −4.42814817589340107113349121722, −4.06893392803200505466540844600, −3.11113211742965305440285355430, −2.50600794793871457785764123029, −1.67048625991240880162090205477, 0,
1.67048625991240880162090205477, 2.50600794793871457785764123029, 3.11113211742965305440285355430, 4.06893392803200505466540844600, 4.42814817589340107113349121722, 5.25394383625954075823188557944, 5.87122648860558932637535318500, 6.36400121037507245629880474392, 7.11786914936833242200876562192
