| L(s) = 1 | − 6·11-s + 8·19-s + 6·29-s + 10·31-s − 6·41-s + 13·49-s − 6·59-s − 26·61-s + 24·71-s − 22·79-s + 12·89-s − 30·101-s − 4·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.80·11-s + 1.83·19-s + 1.11·29-s + 1.79·31-s − 0.937·41-s + 13/7·49-s − 0.781·59-s − 3.32·61-s + 2.84·71-s − 2.47·79-s + 1.27·89-s − 2.98·101-s − 0.383·109-s + 5/11·121-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 + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.549002832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.549002832\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
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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
−8.020447453028799337353840311442, −7.66826553467036677639334793517, −7.49594367466110558551585549445, −6.92856319602506406522801114572, −6.63623637002773688387976756038, −6.34053647716306770695072107889, −5.84958523982413212493460525913, −5.38906709466205995081866338723, −5.34423240662575534142109088982, −4.86665600813687611258936859091, −4.58830202383528325011295839015, −4.14398147801413692709452445638, −3.68124625247560916230419077705, −3.00256914967711954345464833986, −2.95317231329457203333309748326, −2.64089445190707309823216575437, −2.11576947482009928898836048946, −1.33398030643492320724505886367, −1.09345759444443759037937674993, −0.31877350296695350286436707115,
0.31877350296695350286436707115, 1.09345759444443759037937674993, 1.33398030643492320724505886367, 2.11576947482009928898836048946, 2.64089445190707309823216575437, 2.95317231329457203333309748326, 3.00256914967711954345464833986, 3.68124625247560916230419077705, 4.14398147801413692709452445638, 4.58830202383528325011295839015, 4.86665600813687611258936859091, 5.34423240662575534142109088982, 5.38906709466205995081866338723, 5.84958523982413212493460525913, 6.34053647716306770695072107889, 6.63623637002773688387976756038, 6.92856319602506406522801114572, 7.49594367466110558551585549445, 7.66826553467036677639334793517, 8.020447453028799337353840311442
