L(s) = 1 | + 2·3-s + 3·9-s − 16·19-s − 8·25-s + 4·27-s + 8·31-s + 8·37-s − 8·47-s + 12·53-s − 32·57-s − 16·59-s − 16·75-s + 5·81-s − 24·83-s + 16·93-s − 8·103-s − 40·109-s + 16·111-s − 4·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s − 16·141-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 3.67·19-s − 8/5·25-s + 0.769·27-s + 1.43·31-s + 1.31·37-s − 1.16·47-s + 1.64·53-s − 4.23·57-s − 2.08·59-s − 1.84·75-s + 5/9·81-s − 2.63·83-s + 1.65·93-s − 0.788·103-s − 3.83·109-s + 1.51·111-s − 0.376·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.34·141-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
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
−7.60706701343412120187420858530, −7.36996555348267189360297425382, −6.67707781267495867352390317626, −6.50493416827160791433688022995, −6.32461983059197542082826594353, −6.06254060858610539765519137491, −5.32657740901729022209059391577, −5.20094872310647261998469235986, −4.45127307830515547115496387995, −4.25099612002853895132000788890, −3.97613926566938042081398599502, −3.94866147183485483451458537503, −3.04388701822641046890994418298, −2.79588024041744565436390736693, −2.29248202909747841308163648409, −2.25572800064706429160844167719, −1.48343602033940462313512299830, −1.27530079570322944130771380821, 0, 0,
1.27530079570322944130771380821, 1.48343602033940462313512299830, 2.25572800064706429160844167719, 2.29248202909747841308163648409, 2.79588024041744565436390736693, 3.04388701822641046890994418298, 3.94866147183485483451458537503, 3.97613926566938042081398599502, 4.25099612002853895132000788890, 4.45127307830515547115496387995, 5.20094872310647261998469235986, 5.32657740901729022209059391577, 6.06254060858610539765519137491, 6.32461983059197542082826594353, 6.50493416827160791433688022995, 6.67707781267495867352390317626, 7.36996555348267189360297425382, 7.60706701343412120187420858530