| L(s) = 1 | + 2·2-s + 2·4-s + 2·7-s + 2·9-s + 4·14-s − 4·16-s + 4·18-s + 16·23-s − 25-s + 4·28-s − 16·31-s − 8·32-s + 4·36-s − 4·41-s + 32·46-s + 16·47-s + 3·49-s − 2·50-s − 32·62-s + 4·63-s − 8·64-s + 4·71-s + 8·73-s − 4·79-s − 5·81-s − 8·82-s + 4·89-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.755·7-s + 2/3·9-s + 1.06·14-s − 16-s + 0.942·18-s + 3.33·23-s − 1/5·25-s + 0.755·28-s − 2.87·31-s − 1.41·32-s + 2/3·36-s − 0.624·41-s + 4.71·46-s + 2.33·47-s + 3/7·49-s − 0.282·50-s − 4.06·62-s + 0.503·63-s − 64-s + 0.474·71-s + 0.936·73-s − 0.450·79-s − 5/9·81-s − 0.883·82-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.397627232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.397627232\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 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
−12.38376736999930324333143358773, −11.73767487249266841927131091493, −11.00254037234164646613706029393, −10.98609206501897297052335205385, −10.63935191835904951259489965382, −9.574246572956148022134268869640, −9.262673311020243056479973776772, −8.846187386583447001354144671029, −8.252367082791596531678496744475, −7.36047886773411951612778996237, −7.09706859958992277369656938957, −6.80055759671756746949027070232, −5.81488400434806022753942708023, −5.29386791472122641693873848612, −5.11550649019364488951853432169, −4.32555764497868044400610701493, −3.85272602840362160378447277006, −3.15102136648621166249677643412, −2.39355991455377212520811583508, −1.36852156903759209297522432387,
1.36852156903759209297522432387, 2.39355991455377212520811583508, 3.15102136648621166249677643412, 3.85272602840362160378447277006, 4.32555764497868044400610701493, 5.11550649019364488951853432169, 5.29386791472122641693873848612, 5.81488400434806022753942708023, 6.80055759671756746949027070232, 7.09706859958992277369656938957, 7.36047886773411951612778996237, 8.252367082791596531678496744475, 8.846187386583447001354144671029, 9.262673311020243056479973776772, 9.574246572956148022134268869640, 10.63935191835904951259489965382, 10.98609206501897297052335205385, 11.00254037234164646613706029393, 11.73767487249266841927131091493, 12.38376736999930324333143358773
