| L(s) = 1 | − 3·7-s + 2·13-s + 25-s + 3·31-s − 2·37-s + 3·43-s + 5·49-s + 61-s + 3·67-s + 2·73-s − 6·91-s + 2·97-s − 3·103-s + 2·109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s − 3·175-s + ⋯ |
| L(s) = 1 | − 3·7-s + 2·13-s + 25-s + 3·31-s − 2·37-s + 3·43-s + 5·49-s + 61-s + 3·67-s + 2·73-s − 6·91-s + 2·97-s − 3·103-s + 2·109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s − 3·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.236836756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236836756\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−8.724462949810673403297307668936, −8.702877284728050058747689794089, −8.103745366275860782080250283309, −7.897092670168725190807200455329, −7.07239592447533887768093004535, −6.87313905309341377822002747553, −6.55707343940855613197475359219, −6.41855240996943048222842169617, −5.97047825872641130853218933187, −5.68886278334389292373831424057, −5.21697786526327794730513257427, −4.61328014363655551597136191387, −4.01775139991821207683535937787, −3.80603310195016664122215192452, −3.30592738899883602610862650808, −3.17965225083913818140546369184, −2.52842567183140256887380358063, −2.24074128708659151087413854058, −0.923950363691241952173422111122, −0.882913938507464484612179626193,
0.882913938507464484612179626193, 0.923950363691241952173422111122, 2.24074128708659151087413854058, 2.52842567183140256887380358063, 3.17965225083913818140546369184, 3.30592738899883602610862650808, 3.80603310195016664122215192452, 4.01775139991821207683535937787, 4.61328014363655551597136191387, 5.21697786526327794730513257427, 5.68886278334389292373831424057, 5.97047825872641130853218933187, 6.41855240996943048222842169617, 6.55707343940855613197475359219, 6.87313905309341377822002747553, 7.07239592447533887768093004535, 7.897092670168725190807200455329, 8.103745366275860782080250283309, 8.702877284728050058747689794089, 8.724462949810673403297307668936
