| L(s) = 1 | − 9-s − 8·19-s + 12·29-s + 20·41-s − 2·49-s + 12·61-s − 32·79-s + 81-s − 4·89-s − 28·101-s + 20·109-s − 22·121-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 + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.83·19-s + 2.22·29-s + 3.12·41-s − 2/7·49-s + 1.53·61-s − 3.60·79-s + 1/9·81-s − 0.423·89-s − 2.78·101-s + 1.91·109-s − 2·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.564258266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564258266\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−11.18524216341932078040356757274, −10.46690162467379233142421946017, −10.03990665123640350444670335975, −9.638606927349785461511499718977, −9.078807256848167901194755803226, −8.623172100411390454070304948476, −8.287448057278562022482531391962, −7.996633054832418871659441165344, −7.18100296939035834821071657271, −6.93180134066851249310875348524, −6.29720101906815292738912775970, −5.95738066798228525721165691884, −5.49193846641471260402383809720, −4.70067279448851147077453764998, −4.28788880917770057848516441547, −3.96788115327728612406976224596, −2.79134621649960698037904359785, −2.73977795103903345474604748271, −1.78067278372300495508591919348, −0.70899130635208314787033985535,
0.70899130635208314787033985535, 1.78067278372300495508591919348, 2.73977795103903345474604748271, 2.79134621649960698037904359785, 3.96788115327728612406976224596, 4.28788880917770057848516441547, 4.70067279448851147077453764998, 5.49193846641471260402383809720, 5.95738066798228525721165691884, 6.29720101906815292738912775970, 6.93180134066851249310875348524, 7.18100296939035834821071657271, 7.996633054832418871659441165344, 8.287448057278562022482531391962, 8.623172100411390454070304948476, 9.078807256848167901194755803226, 9.638606927349785461511499718977, 10.03990665123640350444670335975, 10.46690162467379233142421946017, 11.18524216341932078040356757274
