| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 11-s − 13-s + 16-s − 6·17-s + 20-s − 22-s + 4·23-s + 25-s − 26-s − 2·29-s + 32-s − 6·34-s + 6·37-s + 40-s + 10·41-s + 8·43-s − 44-s + 4·46-s − 12·47-s − 7·49-s + 50-s − 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.371·29-s + 0.176·32-s − 1.02·34-s + 0.986·37-s + 0.158·40-s + 1.56·41-s + 1.21·43-s − 0.150·44-s + 0.589·46-s − 1.75·47-s − 49-s + 0.141·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.513486663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.513486663\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.22222176556480, −15.64248848502790, −15.03395764881017, −14.59608069054544, −14.02048564791681, −13.35199713133801, −12.85882453674622, −12.67369187672488, −11.57781876036396, −11.23414155705999, −10.72221041694435, −9.948063763895229, −9.354399170269526, −8.773358865600851, −7.967470809364709, −7.308722869052001, −6.668109386348427, −6.106680728241059, −5.426204034111585, −4.715420233671376, −4.237393587381543, −3.276114256461112, −2.512308697202970, −1.955721059448718, −0.7378188604140371,
0.7378188604140371, 1.955721059448718, 2.512308697202970, 3.276114256461112, 4.237393587381543, 4.715420233671376, 5.426204034111585, 6.106680728241059, 6.668109386348427, 7.308722869052001, 7.967470809364709, 8.773358865600851, 9.354399170269526, 9.948063763895229, 10.72221041694435, 11.23414155705999, 11.57781876036396, 12.67369187672488, 12.85882453674622, 13.35199713133801, 14.02048564791681, 14.59608069054544, 15.03395764881017, 15.64248848502790, 16.22222176556480
