L(s) = 1 | − 4-s − 9-s − 6·11-s + 16-s + 10·19-s + 8·29-s − 6·31-s + 36-s + 4·41-s + 6·44-s + 13·49-s + 16·59-s − 4·61-s − 64-s − 12·71-s − 10·76-s + 30·79-s + 81-s − 4·89-s + 6·99-s + 10·101-s + 18·109-s − 8·116-s + 5·121-s + 6·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1/3·9-s − 1.80·11-s + 1/4·16-s + 2.29·19-s + 1.48·29-s − 1.07·31-s + 1/6·36-s + 0.624·41-s + 0.904·44-s + 13/7·49-s + 2.08·59-s − 0.512·61-s − 1/8·64-s − 1.42·71-s − 1.14·76-s + 3.37·79-s + 1/9·81-s − 0.423·89-s + 0.603·99-s + 0.995·101-s + 1.72·109-s − 0.742·116-s + 5/11·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.898810612\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.898810612\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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
−9.167407619902233046494222138902, −8.750665164041458623983831845462, −8.183956694854301605070786512292, −7.990158316841204488947389683822, −7.66821475429571238332984014363, −7.17862736801945245626673622924, −7.02104696732293284844404007740, −6.34975667134583752122398761345, −5.76003013289323366722328892513, −5.43143616327414868446114459656, −5.40710695070715015431231173032, −4.68388897049237560413442757728, −4.58256089390680482103131667670, −3.63809958627429807223981996121, −3.52363024777818826798405531636, −2.81238748262843188628141558196, −2.58568942522214367678035731820, −1.95033174118508858370398304990, −0.988243288867752142068473903686, −0.56614619185317674264079813213,
0.56614619185317674264079813213, 0.988243288867752142068473903686, 1.95033174118508858370398304990, 2.58568942522214367678035731820, 2.81238748262843188628141558196, 3.52363024777818826798405531636, 3.63809958627429807223981996121, 4.58256089390680482103131667670, 4.68388897049237560413442757728, 5.40710695070715015431231173032, 5.43143616327414868446114459656, 5.76003013289323366722328892513, 6.34975667134583752122398761345, 7.02104696732293284844404007740, 7.17862736801945245626673622924, 7.66821475429571238332984014363, 7.990158316841204488947389683822, 8.183956694854301605070786512292, 8.750665164041458623983831845462, 9.167407619902233046494222138902