| L(s) = 1 | + 2·3-s + 3·9-s − 25-s + 4·27-s − 2·49-s − 2·75-s + 5·81-s + 4·83-s − 4·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2·3-s + 3·9-s − 25-s + 4·27-s − 2·49-s − 2·75-s + 5·81-s + 4·83-s − 4·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.421594625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.421594625\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$ | \( ( 1 - T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.835921075437594042301058712956, −8.412680372086298022881127664970, −8.131938945560888023330069863872, −7.80514311241463874765492646872, −7.60318077070547455869941813583, −7.19254386962222246612720149817, −6.56928963257293553817324828523, −6.55231431953066758657640736902, −6.05160993898078168817711921146, −5.20534042094722013275398633718, −5.15638976292757890536125913529, −4.51165860115946266448878954980, −4.08044838945573989754683513699, −3.87090831760038284415997348481, −3.19261171336065580944605673074, −3.14521254806142917723017888224, −2.49481314600376070309051329062, −1.95783286372025922816172437093, −1.74644537100122972964930678902, −0.968726264705978879742969224863,
0.968726264705978879742969224863, 1.74644537100122972964930678902, 1.95783286372025922816172437093, 2.49481314600376070309051329062, 3.14521254806142917723017888224, 3.19261171336065580944605673074, 3.87090831760038284415997348481, 4.08044838945573989754683513699, 4.51165860115946266448878954980, 5.15638976292757890536125913529, 5.20534042094722013275398633718, 6.05160993898078168817711921146, 6.55231431953066758657640736902, 6.56928963257293553817324828523, 7.19254386962222246612720149817, 7.60318077070547455869941813583, 7.80514311241463874765492646872, 8.131938945560888023330069863872, 8.412680372086298022881127664970, 8.835921075437594042301058712956
