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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.239841016\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239841016\) |
\(L(\frac{3}{2})\) |
|
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
−9.027889241861959885587486173493, −8.095145369001344704414318416768, −7.81612587206599255781830128993, −7.28500933842082239871001488842, −7.23130589566269958221591522735, −6.51288879626561545254674699794, −6.28633067041956117914708403611, −5.82105543763721700221125087307, −5.81690269783928820787763513248, −5.06493257573387129424754687241, −4.83230405348179457841508999607, −4.18252272051402475855765849818, −4.12770966459033456279726101144, −3.52216754910067126398048397586, −3.20983558407158514714468589104, −2.36712647095184738611668133738, −2.33005479422131672625401914617, −1.79099712360027512581802621209, −1.05011336400021207754614972501, −0.32704372559597997359209553918,
0.32704372559597997359209553918, 1.05011336400021207754614972501, 1.79099712360027512581802621209, 2.33005479422131672625401914617, 2.36712647095184738611668133738, 3.20983558407158514714468589104, 3.52216754910067126398048397586, 4.12770966459033456279726101144, 4.18252272051402475855765849818, 4.83230405348179457841508999607, 5.06493257573387129424754687241, 5.81690269783928820787763513248, 5.82105543763721700221125087307, 6.28633067041956117914708403611, 6.51288879626561545254674699794, 7.23130589566269958221591522735, 7.28500933842082239871001488842, 7.81612587206599255781830128993, 8.095145369001344704414318416768, 9.027889241861959885587486173493