| L(s) = 1 | − 4-s − 9-s − 10·11-s + 16-s + 6·19-s + 2·29-s + 36-s − 6·41-s + 10·44-s + 13·49-s + 12·59-s + 8·61-s − 64-s − 4·71-s − 6·76-s + 14·79-s + 81-s + 4·89-s + 10·99-s + 12·101-s + 8·109-s − 2·116-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 3.01·11-s + 1/4·16-s + 1.37·19-s + 0.371·29-s + 1/6·36-s − 0.937·41-s + 1.50·44-s + 13/7·49-s + 1.56·59-s + 1.02·61-s − 1/8·64-s − 0.474·71-s − 0.688·76-s + 1.57·79-s + 1/9·81-s + 0.423·89-s + 1.00·99-s + 1.19·101-s + 0.766·109-s − 0.185·116-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.552301857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.552301857\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−8.506839815747948566509869718703, −8.445576083337445954737910750091, −8.052171857764586033046993988303, −7.77913578816492253641195084040, −7.30494089949458958508659379601, −7.09425810597550696010440863856, −6.65679607103381192968213220374, −5.83294016260348288623105060723, −5.72182109306932713930736731536, −5.32861137058530735873395623964, −5.12327723093280251311366919856, −4.65812481894219249691559500651, −4.26764837391147877667386650946, −3.55574248121893819982329163703, −3.15885636741418453288508208255, −2.91273114122607476580467902353, −2.20260587120957763228065576959, −2.06034695100097807851896561480, −0.839845229145255697862272635621, −0.49956069520283870431519432916,
0.49956069520283870431519432916, 0.839845229145255697862272635621, 2.06034695100097807851896561480, 2.20260587120957763228065576959, 2.91273114122607476580467902353, 3.15885636741418453288508208255, 3.55574248121893819982329163703, 4.26764837391147877667386650946, 4.65812481894219249691559500651, 5.12327723093280251311366919856, 5.32861137058530735873395623964, 5.72182109306932713930736731536, 5.83294016260348288623105060723, 6.65679607103381192968213220374, 7.09425810597550696010440863856, 7.30494089949458958508659379601, 7.77913578816492253641195084040, 8.052171857764586033046993988303, 8.445576083337445954737910750091, 8.506839815747948566509869718703
