| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·9-s + 13-s + 5·16-s + 17-s − 4·18-s − 2·26-s − 29-s − 6·32-s − 2·34-s + 6·36-s + 37-s − 41-s + 2·49-s + 3·52-s + 53-s + 2·58-s − 61-s + 7·64-s + 3·68-s − 8·72-s + 73-s − 2·74-s + 3·81-s + 2·82-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·9-s + 13-s + 5·16-s + 17-s − 4·18-s − 2·26-s − 29-s − 6·32-s − 2·34-s + 6·36-s + 37-s − 41-s + 2·49-s + 3·52-s + 53-s + 2·58-s − 61-s + 7·64-s + 3·68-s − 8·72-s + 73-s − 2·74-s + 3·81-s + 2·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7760475952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7760475952\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 17 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 97 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.350565196064229383003616765204, −8.944191339179056758140172353089, −8.522717546213237955688312177516, −8.261927763184136426272610755036, −7.67860420595458069790843664791, −7.50358998789650268085332253260, −7.23524900688945338908853768260, −6.82384297810429551325557449673, −6.37250204457914593014868612169, −6.11307589142641868293019243987, −5.41644235033860386479727801822, −5.36046846446162242401468695755, −4.21411698516904211507860939868, −4.15073267321779786561828287260, −3.31625537045838323918266477847, −3.22031490164283915784657848588, −2.13213772147647055134379176861, −2.04785826340634829589181124630, −1.12162084521037929633987422510, −1.06887600902116701933945795472,
1.06887600902116701933945795472, 1.12162084521037929633987422510, 2.04785826340634829589181124630, 2.13213772147647055134379176861, 3.22031490164283915784657848588, 3.31625537045838323918266477847, 4.15073267321779786561828287260, 4.21411698516904211507860939868, 5.36046846446162242401468695755, 5.41644235033860386479727801822, 6.11307589142641868293019243987, 6.37250204457914593014868612169, 6.82384297810429551325557449673, 7.23524900688945338908853768260, 7.50358998789650268085332253260, 7.67860420595458069790843664791, 8.261927763184136426272610755036, 8.522717546213237955688312177516, 8.944191339179056758140172353089, 9.350565196064229383003616765204
