| L(s) = 1 | + 3·4-s − 9-s − 12·11-s + 5·16-s − 2·19-s − 8·29-s − 3·36-s − 36·44-s + 10·49-s − 24·59-s + 4·61-s + 3·64-s + 32·71-s − 6·76-s − 16·79-s + 81-s + 12·99-s − 36·101-s + 12·109-s − 24·116-s + 86·121-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + 149-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1/3·9-s − 3.61·11-s + 5/4·16-s − 0.458·19-s − 1.48·29-s − 1/2·36-s − 5.42·44-s + 10/7·49-s − 3.12·59-s + 0.512·61-s + 3/8·64-s + 3.79·71-s − 0.688·76-s − 1.80·79-s + 1/9·81-s + 1.20·99-s − 3.58·101-s + 1.14·109-s − 2.22·116-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2030625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2030625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.278675886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.278675886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−10.15479965577495699832978911352, −9.351471951766441361473361864459, −8.947217501679964739528366579863, −8.267226605916544117749299181595, −7.982338082391660254885536409864, −7.75331431266845502753519158350, −7.46515482458678458890735538015, −6.89114476296741186754401743500, −6.63237319163602887396808023249, −5.79072266519328964800501616243, −5.73321900409454189265297261868, −5.32341880375655928565008582686, −4.88347127068724331216009089231, −4.25081530838063350370727613507, −3.51153831935527777392753469217, −2.84084566744910887499582391347, −2.75537441579377559796921450211, −2.17363756794423100136012347804, −1.75986914925693654187063371334, −0.40487527694469688881034360798,
0.40487527694469688881034360798, 1.75986914925693654187063371334, 2.17363756794423100136012347804, 2.75537441579377559796921450211, 2.84084566744910887499582391347, 3.51153831935527777392753469217, 4.25081530838063350370727613507, 4.88347127068724331216009089231, 5.32341880375655928565008582686, 5.73321900409454189265297261868, 5.79072266519328964800501616243, 6.63237319163602887396808023249, 6.89114476296741186754401743500, 7.46515482458678458890735538015, 7.75331431266845502753519158350, 7.982338082391660254885536409864, 8.267226605916544117749299181595, 8.947217501679964739528366579863, 9.351471951766441361473361864459, 10.15479965577495699832978911352
