| L(s) = 1 | + 2·7-s + 4·11-s + 4·13-s + 8·17-s + 2·19-s + 4·23-s − 2·25-s − 4·29-s + 8·31-s − 12·37-s + 4·41-s − 8·43-s + 4·47-s + 3·49-s + 12·53-s + 8·59-s − 12·61-s + 16·71-s + 20·73-s + 8·77-s + 24·79-s − 20·83-s − 12·89-s + 8·91-s − 12·97-s − 8·101-s − 16·107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20·11-s + 1.10·13-s + 1.94·17-s + 0.458·19-s + 0.834·23-s − 2/5·25-s − 0.742·29-s + 1.43·31-s − 1.97·37-s + 0.624·41-s − 1.21·43-s + 0.583·47-s + 3/7·49-s + 1.64·53-s + 1.04·59-s − 1.53·61-s + 1.89·71-s + 2.34·73-s + 0.911·77-s + 2.70·79-s − 2.19·83-s − 1.27·89-s + 0.838·91-s − 1.21·97-s − 0.796·101-s − 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.165267217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.165267217\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 234 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.88142458468792024348663577290, −7.68576171640838641716617819664, −6.98522082407248989967374505284, −6.90046403717253718927776892078, −6.57575646943681941631616328515, −6.24552459612375265420994281906, −5.54395625578144382551682015860, −5.49632166475765640179097435652, −5.30630480734659459205707917232, −4.84382735532714466020125342564, −4.20470924509996335250549213574, −4.00387119234400367666487843915, −3.57865061318861412133674264503, −3.43549806058504962570421027651, −2.84853032749248941539428549305, −2.43826742807181675283275948670, −1.66351920986472524263283125739, −1.53898920487762721399694196299, −0.991213546752919718324897353651, −0.66667112308191374579033810868,
0.66667112308191374579033810868, 0.991213546752919718324897353651, 1.53898920487762721399694196299, 1.66351920986472524263283125739, 2.43826742807181675283275948670, 2.84853032749248941539428549305, 3.43549806058504962570421027651, 3.57865061318861412133674264503, 4.00387119234400367666487843915, 4.20470924509996335250549213574, 4.84382735532714466020125342564, 5.30630480734659459205707917232, 5.49632166475765640179097435652, 5.54395625578144382551682015860, 6.24552459612375265420994281906, 6.57575646943681941631616328515, 6.90046403717253718927776892078, 6.98522082407248989967374505284, 7.68576171640838641716617819664, 7.88142458468792024348663577290
