| L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s − 2·9-s − 3·14-s + 16-s − 2·17-s − 2·18-s − 4·23-s + 25-s − 3·28-s − 12·31-s + 32-s − 2·34-s − 2·36-s + 4·41-s − 4·46-s + 2·47-s + 2·49-s + 50-s − 3·56-s − 12·62-s + 6·63-s + 64-s − 2·68-s − 7·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 2/3·9-s − 0.801·14-s + 1/4·16-s − 0.485·17-s − 0.471·18-s − 0.834·23-s + 1/5·25-s − 0.566·28-s − 2.15·31-s + 0.176·32-s − 0.342·34-s − 1/3·36-s + 0.624·41-s − 0.589·46-s + 0.291·47-s + 2/7·49-s + 0.141·50-s − 0.400·56-s − 1.52·62-s + 0.755·63-s + 1/8·64-s − 0.242·68-s − 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.887941105872476366072573708727, −8.787633502628813252101191603226, −7.83159636296615772660520701245, −7.59847424584424245976938225648, −6.79450541688216545276976395983, −6.62550670548588926941061615798, −5.85881558798639804438121874024, −5.64074499004200893628060864908, −5.03674051889363540218177633601, −4.10862632850436663973882287719, −3.88581998433598041695051543132, −3.05314363152948468440858942308, −2.62009072778936741348427316055, −1.69170566717355238264052962835, 0,
1.69170566717355238264052962835, 2.62009072778936741348427316055, 3.05314363152948468440858942308, 3.88581998433598041695051543132, 4.10862632850436663973882287719, 5.03674051889363540218177633601, 5.64074499004200893628060864908, 5.85881558798639804438121874024, 6.62550670548588926941061615798, 6.79450541688216545276976395983, 7.59847424584424245976938225648, 7.83159636296615772660520701245, 8.787633502628813252101191603226, 8.887941105872476366072573708727
