| L(s) = 1 | − 2·2-s − 4·3-s + 4·4-s + 5·5-s + 8·6-s + 7·7-s − 8·8-s − 11·9-s − 10·10-s − 11·11-s − 16·12-s + 6·13-s − 14·14-s − 20·15-s + 16·16-s − 6·17-s + 22·18-s + 84·19-s + 20·20-s − 28·21-s + 22·22-s − 112·23-s + 32·24-s + 25·25-s − 12·26-s + 152·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.769·3-s + 1/2·4-s + 0.447·5-s + 0.544·6-s + 0.377·7-s − 0.353·8-s − 0.407·9-s − 0.316·10-s − 0.301·11-s − 0.384·12-s + 0.128·13-s − 0.267·14-s − 0.344·15-s + 1/4·16-s − 0.0856·17-s + 0.288·18-s + 1.01·19-s + 0.223·20-s − 0.290·21-s + 0.213·22-s − 1.01·23-s + 0.272·24-s + 1/5·25-s − 0.0905·26-s + 1.08·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9795715139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9795715139\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 84 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 34 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 - 286 T + p^{3} T^{2} \) |
| 41 | \( 1 + 350 T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 528 T + p^{3} T^{2} \) |
| 53 | \( 1 - 654 T + p^{3} T^{2} \) |
| 59 | \( 1 + 380 T + p^{3} T^{2} \) |
| 61 | \( 1 + 538 T + p^{3} T^{2} \) |
| 67 | \( 1 + 156 T + p^{3} T^{2} \) |
| 71 | \( 1 - 608 T + p^{3} T^{2} \) |
| 73 | \( 1 + 798 T + p^{3} T^{2} \) |
| 79 | \( 1 + 640 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1044 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1386 T + p^{3} T^{2} \) |
| 97 | \( 1 - 338 T + p^{3} T^{2} \) |
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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
−10.02131503063033310418796053702, −9.131867116593489233069877969152, −8.235963539473113728131164496893, −7.42660035956371925177605111915, −6.31272245162636605572801765705, −5.68106243295467796435837532839, −4.74447521439272910279823850867, −3.17311357495241699163425537906, −1.91954785443575147540160199465, −0.64135309419595621129443389055,
0.64135309419595621129443389055, 1.91954785443575147540160199465, 3.17311357495241699163425537906, 4.74447521439272910279823850867, 5.68106243295467796435837532839, 6.31272245162636605572801765705, 7.42660035956371925177605111915, 8.235963539473113728131164496893, 9.131867116593489233069877969152, 10.02131503063033310418796053702
