L(s) = 1 | − 8·19-s − 12·29-s − 16·31-s + 12·41-s − 2·49-s − 20·61-s + 16·79-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 + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | − 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 + 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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3578326089\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3578326089\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 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
−8.851360859386031912399177039006, −8.377956340485385274708632349968, −7.82418348307318354353349829140, −7.61771289620363914235698612092, −7.46178076909973119732509138908, −6.84408241163408889096336058762, −6.54567292886018637274414191838, −5.98272706848108211284266755523, −5.86604763262579823906092225746, −5.44009592867965687463818999012, −4.85538987404167509287345743113, −4.62332084524881264141357429385, −3.95914062930162466804642858021, −3.69961591616037270434502724535, −3.47182278170122975687984566840, −2.62470554339778977087531446619, −2.18276099877530284891619454517, −1.87179097192214152748665309522, −1.25912666591265772788830563511, −0.17443951772062310226240414068,
0.17443951772062310226240414068, 1.25912666591265772788830563511, 1.87179097192214152748665309522, 2.18276099877530284891619454517, 2.62470554339778977087531446619, 3.47182278170122975687984566840, 3.69961591616037270434502724535, 3.95914062930162466804642858021, 4.62332084524881264141357429385, 4.85538987404167509287345743113, 5.44009592867965687463818999012, 5.86604763262579823906092225746, 5.98272706848108211284266755523, 6.54567292886018637274414191838, 6.84408241163408889096336058762, 7.46178076909973119732509138908, 7.61771289620363914235698612092, 7.82418348307318354353349829140, 8.377956340485385274708632349968, 8.851360859386031912399177039006