| L(s) = 1 | + 4-s − 7-s + 4·9-s − 6·11-s + 16-s − 6·23-s − 28-s + 6·29-s + 4·36-s − 4·37-s + 2·43-s − 6·44-s + 49-s − 6·53-s − 4·63-s + 64-s + 8·67-s − 18·71-s + 6·77-s − 2·79-s + 7·81-s − 6·92-s − 24·99-s + 24·107-s + 22·109-s − 112-s + 24·113-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.377·7-s + 4/3·9-s − 1.80·11-s + 1/4·16-s − 1.25·23-s − 0.188·28-s + 1.11·29-s + 2/3·36-s − 0.657·37-s + 0.304·43-s − 0.904·44-s + 1/7·49-s − 0.824·53-s − 0.503·63-s + 1/8·64-s + 0.977·67-s − 2.13·71-s + 0.683·77-s − 0.225·79-s + 7/9·81-s − 0.625·92-s − 2.41·99-s + 2.32·107-s + 2.10·109-s − 0.0944·112-s + 2.25·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.836049899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.836049899\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.224908148167364547660793016807, −7.61448595613393286757703217481, −7.34439112106834870971810083750, −7.04440581801966531658283867713, −6.40576593098957435909144341550, −5.97707360436401213740399304111, −5.62721286743517268175596275421, −4.87655970304355007216450131088, −4.61582343222263333455122657825, −4.06261170590926391034271732868, −3.28878086518254268163990412705, −2.95912727420180521746052455197, −2.14919988508205179629237745406, −1.77207983872898332466198564186, −0.62587925868655281423821403195,
0.62587925868655281423821403195, 1.77207983872898332466198564186, 2.14919988508205179629237745406, 2.95912727420180521746052455197, 3.28878086518254268163990412705, 4.06261170590926391034271732868, 4.61582343222263333455122657825, 4.87655970304355007216450131088, 5.62721286743517268175596275421, 5.97707360436401213740399304111, 6.40576593098957435909144341550, 7.04440581801966531658283867713, 7.34439112106834870971810083750, 7.61448595613393286757703217481, 8.224908148167364547660793016807
