| L(s) = 1 | − 9-s + 8·19-s − 12·29-s − 16·31-s − 12·41-s − 2·49-s + 20·61-s + 16·79-s + 81-s − 36·89-s − 36·101-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 + 22·169-s − 8·171-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.83·19-s − 2.22·29-s − 2.87·31-s − 1.87·41-s − 2/7·49-s + 2.56·61-s + 1.80·79-s + 1/9·81-s − 3.81·89-s − 3.58·101-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 + 1.69·169-s − 0.611·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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
| \(L(1)\) |
\(\approx\) |
\(0.5617778566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5617778566\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 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
−8.537395137449503963203691087052, −7.966263376202387062752261235190, −7.80235107634565824670375211405, −7.29191628020203835954349274503, −7.11416967298402658014540010980, −6.75285843438458058155856004519, −6.38421865062114281646091783512, −5.63252409518542391525213956678, −5.46589143034714443083993940067, −5.25694533812978407052290526532, −5.13574751213283209023778066655, −4.06876884228442224896906616215, −3.98344400497615501100868244564, −3.58157274485400946865577139421, −3.17936051963050208815382389477, −2.68190670110707957504831908173, −2.12628002976967149965764387077, −1.59155952422769810999849494925, −1.26626555704771791504293917164, −0.20269585753409633530984875096,
0.20269585753409633530984875096, 1.26626555704771791504293917164, 1.59155952422769810999849494925, 2.12628002976967149965764387077, 2.68190670110707957504831908173, 3.17936051963050208815382389477, 3.58157274485400946865577139421, 3.98344400497615501100868244564, 4.06876884228442224896906616215, 5.13574751213283209023778066655, 5.25694533812978407052290526532, 5.46589143034714443083993940067, 5.63252409518542391525213956678, 6.38421865062114281646091783512, 6.75285843438458058155856004519, 7.11416967298402658014540010980, 7.29191628020203835954349274503, 7.80235107634565824670375211405, 7.966263376202387062752261235190, 8.537395137449503963203691087052
