| L(s) = 1 | + 2·3-s + 2·5-s − 7-s + 3·9-s − 2·11-s + 4·13-s + 4·15-s + 3·17-s + 4·19-s − 2·21-s − 2·23-s + 3·25-s + 4·27-s + 29-s + 3·31-s − 4·33-s − 2·35-s + 9·37-s + 8·39-s + 7·41-s + 6·45-s − 8·47-s − 9·49-s + 6·51-s + 11·53-s − 4·55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.377·7-s + 9-s − 0.603·11-s + 1.10·13-s + 1.03·15-s + 0.727·17-s + 0.917·19-s − 0.436·21-s − 0.417·23-s + 3/5·25-s + 0.769·27-s + 0.185·29-s + 0.538·31-s − 0.696·33-s − 0.338·35-s + 1.47·37-s + 1.28·39-s + 1.09·41-s + 0.894·45-s − 1.16·47-s − 9/7·49-s + 0.840·51-s + 1.51·53-s − 0.539·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.400216219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.400216219\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 90 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 56 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 132 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 116 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 102 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 144 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 190 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 170 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 226 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.277266617868684416563278175221, −8.024489923883684005752256161734, −7.64639972560922797713387958247, −7.49798166321625942112823659404, −6.77541428932317051535252519573, −6.67894870060593962394100505544, −6.04079091317727681557359135647, −6.02289684934293703856018097975, −5.34711075332276694848618076125, −5.27182153989776344005139817517, −4.44719790658571881666587106621, −4.41651864173653427063270264343, −3.63389199898900878820170045706, −3.48836734729989045598260061026, −2.86322783719428630909232653588, −2.79920083663139171802560760543, −2.13340463348737489454252899177, −1.76050307300593083884306030466, −1.09899276835850695300664812720, −0.73011891525904154094942911784,
0.73011891525904154094942911784, 1.09899276835850695300664812720, 1.76050307300593083884306030466, 2.13340463348737489454252899177, 2.79920083663139171802560760543, 2.86322783719428630909232653588, 3.48836734729989045598260061026, 3.63389199898900878820170045706, 4.41651864173653427063270264343, 4.44719790658571881666587106621, 5.27182153989776344005139817517, 5.34711075332276694848618076125, 6.02289684934293703856018097975, 6.04079091317727681557359135647, 6.67894870060593962394100505544, 6.77541428932317051535252519573, 7.49798166321625942112823659404, 7.64639972560922797713387958247, 8.024489923883684005752256161734, 8.277266617868684416563278175221
