| L(s) = 1 | − 5-s − 3·7-s − 2·11-s + 5·13-s − 5·17-s − 7·19-s − 5·23-s − 25-s − 3·29-s − 7·31-s + 3·35-s + 6·37-s − 12·41-s − 8·43-s − 3·47-s + 49-s − 10·53-s + 2·55-s − 14·59-s − 61-s − 5·65-s − 4·67-s + 8·71-s − 7·73-s + 6·77-s + 7·79-s − 25·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 0.603·11-s + 1.38·13-s − 1.21·17-s − 1.60·19-s − 1.04·23-s − 1/5·25-s − 0.557·29-s − 1.25·31-s + 0.507·35-s + 0.986·37-s − 1.87·41-s − 1.21·43-s − 0.437·47-s + 1/7·49-s − 1.37·53-s + 0.269·55-s − 1.82·59-s − 0.128·61-s − 0.620·65-s − 0.488·67-s + 0.949·71-s − 0.819·73-s + 0.683·77-s + 0.787·79-s − 2.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 66 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 88 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 114 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 105 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 162 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 25 T + 314 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 177 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.394158874294007273996782172605, −8.979472142573530964819046455050, −8.727770223738188031122951495364, −8.121269556712013142674166634499, −8.041178318912997030205839562982, −7.44786911605943622831691737688, −6.80774154888078831344379910125, −6.56769978400851266016617686027, −6.08502863571544441518884100940, −6.01212845975460546969550924587, −5.17202413054534468385232123754, −4.74568525600728611458822717237, −4.04015392546546353758577951625, −3.92564790152860563571748785695, −3.24762037428474586875954999195, −2.89236116533902573643406665998, −1.94950947461035593346148136329, −1.67130554709750835854018785293, 0, 0,
1.67130554709750835854018785293, 1.94950947461035593346148136329, 2.89236116533902573643406665998, 3.24762037428474586875954999195, 3.92564790152860563571748785695, 4.04015392546546353758577951625, 4.74568525600728611458822717237, 5.17202413054534468385232123754, 6.01212845975460546969550924587, 6.08502863571544441518884100940, 6.56769978400851266016617686027, 6.80774154888078831344379910125, 7.44786911605943622831691737688, 8.041178318912997030205839562982, 8.121269556712013142674166634499, 8.727770223738188031122951495364, 8.979472142573530964819046455050, 9.394158874294007273996782172605
