| L(s) = 1 | + 18·9-s − 32·11-s + 8·19-s + 484·29-s − 200·31-s − 276·41-s − 470·49-s − 536·59-s + 500·61-s + 1.70e3·71-s − 912·79-s − 405·81-s + 1.45e3·89-s − 576·99-s + 252·101-s − 52·109-s − 1.89e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.03e3·169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 0.877·11-s + 0.0965·19-s + 3.09·29-s − 1.15·31-s − 1.05·41-s − 1.37·49-s − 1.18·59-s + 1.04·61-s + 2.84·71-s − 1.29·79-s − 5/9·81-s + 1.72·89-s − 0.584·99-s + 0.248·101-s − 0.0456·109-s − 1.42·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.468·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.172674360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.172674360\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 470 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4926 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6378 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 242 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 90538 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 138 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 127330 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 207162 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 271510 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 268 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 250 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 423442 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 p T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 684398 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 456 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 955218 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 726 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 73538 T^{2} + p^{6} T^{4} \) |
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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
−10.84647483471407206633882960493, −10.65826032989368437312712902917, −10.23063070210092321225062204636, −9.664062918747706105855953769166, −9.497745797918961760249217369403, −8.501489940770908291551762169589, −8.477831212631703126823508346588, −7.86478149262175088128876572911, −7.37929025720895743040115393577, −6.74282190826172659511437740234, −6.51582678129768632772835033799, −5.82456971045870520194256828519, −4.98514548673724351049126967314, −4.95670295008927289785875435569, −4.19442873377851438843916687330, −3.45748659073392774035359063936, −2.90170012563987265123658771305, −2.18412211548998227946560321303, −1.37921044625640459165007256554, −0.51723925781288068149057793760,
0.51723925781288068149057793760, 1.37921044625640459165007256554, 2.18412211548998227946560321303, 2.90170012563987265123658771305, 3.45748659073392774035359063936, 4.19442873377851438843916687330, 4.95670295008927289785875435569, 4.98514548673724351049126967314, 5.82456971045870520194256828519, 6.51582678129768632772835033799, 6.74282190826172659511437740234, 7.37929025720895743040115393577, 7.86478149262175088128876572911, 8.477831212631703126823508346588, 8.501489940770908291551762169589, 9.497745797918961760249217369403, 9.664062918747706105855953769166, 10.23063070210092321225062204636, 10.65826032989368437312712902917, 10.84647483471407206633882960493
