| L(s) = 1 | − 9-s + 8·11-s − 8·19-s + 20·29-s + 8·31-s + 4·41-s − 2·49-s − 24·59-s + 20·61-s − 8·79-s + 81-s + 12·89-s − 8·99-s − 4·101-s − 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 8·171-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.41·11-s − 1.83·19-s + 3.71·29-s + 1.43·31-s + 0.624·41-s − 2/7·49-s − 3.12·59-s + 2.56·61-s − 0.900·79-s + 1/9·81-s + 1.27·89-s − 0.804·99-s − 0.398·101-s − 0.383·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.611·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.507715849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.507715849\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.584119355078238384327730079764, −8.320759430949247137123979521730, −7.890115889827457421395837682132, −7.26041853257305563852669202783, −6.79937850580809708744760189844, −6.59120736527445318771707287594, −6.43217360295673038649170092071, −6.02923909913756187672126723793, −5.79516916435806824125314969023, −4.94182812551242971974275229238, −4.62109249606848269075016538302, −4.37309331160961629324975021252, −4.17270554082565674402250822869, −3.42960393718799903279700968724, −3.24592859438569201523130937692, −2.49008195327646757761934316800, −2.35383117957665200622541531302, −1.47686538359228622183899420901, −1.14972286763986108183675783794, −0.57561421625850403412550751115,
0.57561421625850403412550751115, 1.14972286763986108183675783794, 1.47686538359228622183899420901, 2.35383117957665200622541531302, 2.49008195327646757761934316800, 3.24592859438569201523130937692, 3.42960393718799903279700968724, 4.17270554082565674402250822869, 4.37309331160961629324975021252, 4.62109249606848269075016538302, 4.94182812551242971974275229238, 5.79516916435806824125314969023, 6.02923909913756187672126723793, 6.43217360295673038649170092071, 6.59120736527445318771707287594, 6.79937850580809708744760189844, 7.26041853257305563852669202783, 7.890115889827457421395837682132, 8.320759430949247137123979521730, 8.584119355078238384327730079764
