| L(s) = 1 | + 5-s + 4·7-s + 6·11-s + 4·13-s − 6·17-s + 6·23-s + 25-s + 2·29-s + 4·35-s + 8·37-s + 10·41-s + 4·43-s − 2·47-s + 9·49-s + 10·53-s + 6·55-s + 12·59-s + 2·61-s + 4·65-s − 8·67-s − 6·73-s + 24·77-s + 4·79-s + 2·83-s − 6·85-s − 14·89-s + 16·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.80·11-s + 1.10·13-s − 1.45·17-s + 1.25·23-s + 1/5·25-s + 0.371·29-s + 0.676·35-s + 1.31·37-s + 1.56·41-s + 0.609·43-s − 0.291·47-s + 9/7·49-s + 1.37·53-s + 0.809·55-s + 1.56·59-s + 0.256·61-s + 0.496·65-s − 0.977·67-s − 0.702·73-s + 2.73·77-s + 0.450·79-s + 0.219·83-s − 0.650·85-s − 1.48·89-s + 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.419443709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.419443709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.88154047812891, −12.31741408176362, −11.69507381619780, −11.38507964546621, −10.95946603560817, −10.87623136189882, −10.00611134529182, −9.422086570740962, −9.026538635885994, −8.599866012360791, −8.428663971144553, −7.600946104309333, −7.129084180867368, −6.628429328727546, −6.204456753813103, −5.713298848108703, −5.113607392220083, −4.521013281865172, −4.065219713482458, −3.874802338245671, −2.762619235930107, −2.405193172683274, −1.588066996185074, −1.223182152654496, −0.7700961952410870,
0.7700961952410870, 1.223182152654496, 1.588066996185074, 2.405193172683274, 2.762619235930107, 3.874802338245671, 4.065219713482458, 4.521013281865172, 5.113607392220083, 5.713298848108703, 6.204456753813103, 6.628429328727546, 7.129084180867368, 7.600946104309333, 8.428663971144553, 8.599866012360791, 9.026538635885994, 9.422086570740962, 10.00611134529182, 10.87623136189882, 10.95946603560817, 11.38507964546621, 11.69507381619780, 12.31741408176362, 12.88154047812891
