L(s) = 1 | + 2-s − 4-s + 3·5-s + 7-s − 3·8-s − 3·9-s + 3·10-s + 4·13-s + 14-s − 16-s − 6·17-s − 3·18-s − 3·20-s + 6·23-s + 4·25-s + 4·26-s − 28-s + 4·29-s − 8·31-s + 5·32-s − 6·34-s + 3·35-s + 3·36-s + 37-s − 9·40-s + 10·41-s − 4·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.34·5-s + 0.377·7-s − 1.06·8-s − 9-s + 0.948·10-s + 1.10·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s − 0.670·20-s + 1.25·23-s + 4/5·25-s + 0.784·26-s − 0.188·28-s + 0.742·29-s − 1.43·31-s + 0.883·32-s − 1.02·34-s + 0.507·35-s + 1/2·36-s + 0.164·37-s − 1.42·40-s + 1.56·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.70929490409057, −14.23946139333000, −13.94901473395772, −13.37205552761277, −13.05977563015135, −12.68912946525413, −11.83529045368181, −11.20281842053816, −10.96764863005356, −10.31400141530275, −9.485840019637426, −9.092758951082921, −8.757250087055542, −8.302287227601251, −7.371068244598056, −6.547618730277065, −6.134484858244694, −5.717494647679395, −5.146815183228025, −4.629524122420375, −3.966829725742751, −3.143326802798808, −2.675183607130749, −1.898922975986664, −1.094459419675251, 0,
1.094459419675251, 1.898922975986664, 2.675183607130749, 3.143326802798808, 3.966829725742751, 4.629524122420375, 5.146815183228025, 5.717494647679395, 6.134484858244694, 6.547618730277065, 7.371068244598056, 8.302287227601251, 8.757250087055542, 9.092758951082921, 9.485840019637426, 10.31400141530275, 10.96764863005356, 11.20281842053816, 11.83529045368181, 12.68912946525413, 13.05977563015135, 13.37205552761277, 13.94901473395772, 14.23946139333000, 14.70929490409057