| L(s) = 1 | + 22·9-s − 40·11-s − 168·19-s − 12·29-s + 448·31-s + 532·41-s − 610·49-s − 56·59-s + 364·61-s − 816·71-s + 96·79-s − 245·81-s + 3.05e3·89-s − 880·99-s + 2.49e3·101-s − 1.80e3·109-s − 1.46e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.65e3·169-s + ⋯ |
| L(s) = 1 | + 0.814·9-s − 1.09·11-s − 2.02·19-s − 0.0768·29-s + 2.59·31-s + 2.02·41-s − 1.77·49-s − 0.123·59-s + 0.764·61-s − 1.36·71-s + 0.136·79-s − 0.336·81-s + 3.63·89-s − 0.893·99-s + 2.45·101-s − 1.58·109-s − 1.09·121-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 − 0.754·169-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
| \(L(2)\) |
\(\approx\) |
\(1.827563439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827563439\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 22 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 610 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1658 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4962 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 84 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 20610 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 86410 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 266 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 65914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 67122 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 163690 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 419050 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 408 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 390542 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1103370 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1526 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1514050 T^{2} + p^{6} T^{4} \) |
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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
−10.81452843163982808038560960039, −10.62925114251768061400764282532, −10.26671805452342738592084242983, −9.827580077384574275833074966703, −9.282833413302734555711824787553, −8.755152502623426094508212705582, −8.245177541893390503204114105373, −7.85400388696147845491978591919, −7.49497566385292140664204160808, −6.76308618910737609690013885051, −6.20042441958520874128415203713, −6.13055827566084910750344003704, −5.06917790250011866328137380557, −4.70673510320637964930287320634, −4.26485984579328047422001422113, −3.59828795094301269537805474975, −2.63935847010885258060933693831, −2.35426515648302918600159376835, −1.37987131608636555704027244616, −0.46738346301242291680060142846,
0.46738346301242291680060142846, 1.37987131608636555704027244616, 2.35426515648302918600159376835, 2.63935847010885258060933693831, 3.59828795094301269537805474975, 4.26485984579328047422001422113, 4.70673510320637964930287320634, 5.06917790250011866328137380557, 6.13055827566084910750344003704, 6.20042441958520874128415203713, 6.76308618910737609690013885051, 7.49497566385292140664204160808, 7.85400388696147845491978591919, 8.245177541893390503204114105373, 8.755152502623426094508212705582, 9.282833413302734555711824787553, 9.827580077384574275833074966703, 10.26671805452342738592084242983, 10.62925114251768061400764282532, 10.81452843163982808038560960039
