L(s) = 1 | − 2·3-s + 2·5-s + 3·9-s + 4·13-s − 4·15-s − 25-s − 4·27-s − 16·31-s − 4·37-s − 8·39-s + 12·41-s + 8·43-s + 6·45-s + 10·49-s + 12·53-s + 8·65-s + 8·71-s + 2·75-s + 16·79-s + 5·81-s − 24·83-s + 28·89-s + 32·93-s − 24·107-s + 8·111-s + 12·117-s − 14·121-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.894·5-s + 9-s + 1.10·13-s − 1.03·15-s − 1/5·25-s − 0.769·27-s − 2.87·31-s − 0.657·37-s − 1.28·39-s + 1.87·41-s + 1.21·43-s + 0.894·45-s + 10/7·49-s + 1.64·53-s + 0.992·65-s + 0.949·71-s + 0.230·75-s + 1.80·79-s + 5/9·81-s − 2.63·83-s + 2.96·89-s + 3.31·93-s − 2.32·107-s + 0.759·111-s + 1.10·117-s − 1.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.066658788\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.066658788\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.130684710735546472698892931101, −8.292700767851739626782211240593, −7.84953048880639867771713975615, −7.40102920453523433446155609914, −7.25606210715840810081145232024, −6.58617006404089407323509436272, −6.55957079198359568851764637453, −5.82704583619405326715213014084, −5.75500582590622337304831424919, −5.38086994955392843423990966407, −5.29548469957176748482197818513, −4.33099300777086362962418548851, −4.25198229439711488115949117055, −3.68435918566779454826221583486, −3.42067582328660837934116204154, −2.49536356185492925617822647012, −2.19837898300958541066231917449, −1.66019413597225531302210505812, −1.07825183516675888762605413422, −0.51242014267888937537038522792,
0.51242014267888937537038522792, 1.07825183516675888762605413422, 1.66019413597225531302210505812, 2.19837898300958541066231917449, 2.49536356185492925617822647012, 3.42067582328660837934116204154, 3.68435918566779454826221583486, 4.25198229439711488115949117055, 4.33099300777086362962418548851, 5.29548469957176748482197818513, 5.38086994955392843423990966407, 5.75500582590622337304831424919, 5.82704583619405326715213014084, 6.55957079198359568851764637453, 6.58617006404089407323509436272, 7.25606210715840810081145232024, 7.40102920453523433446155609914, 7.84953048880639867771713975615, 8.292700767851739626782211240593, 9.130684710735546472698892931101