| L(s) = 1 | − 6·11-s + 10·13-s + 18·23-s + 7·25-s − 4·37-s + 6·47-s + 11·49-s + 6·59-s + 2·61-s + 24·71-s − 4·73-s − 30·83-s − 10·97-s − 24·107-s + 28·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s − 60·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 49·169-s + ⋯ |
| L(s) = 1 | − 1.80·11-s + 2.77·13-s + 3.75·23-s + 7/5·25-s − 0.657·37-s + 0.875·47-s + 11/7·49-s + 0.781·59-s + 0.256·61-s + 2.84·71-s − 0.468·73-s − 3.29·83-s − 1.01·97-s − 2.32·107-s + 2.68·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.01·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.76·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.917234512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.917234512\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−8.465228881280003051578712643536, −8.139653863252550983731007187116, −7.72323171442808617548991375098, −7.14596048291568799543740861323, −6.89850090292395935613614161426, −6.84645472681785116933849835324, −6.24778771839563916872662210302, −5.62516165285240581875011394205, −5.56302419554584969429267005843, −5.22441423566521426453473919341, −4.77871361540091166158362167198, −4.40117098225425295021103046427, −3.77751958887345520929614762693, −3.49281637379518849996451206796, −3.00719446299397611625270487578, −2.73488000083933898891814037984, −2.31659759631235003652655109615, −1.36142583811054545957352483151, −1.10181115834356533336870162408, −0.63295805193117981292190926546,
0.63295805193117981292190926546, 1.10181115834356533336870162408, 1.36142583811054545957352483151, 2.31659759631235003652655109615, 2.73488000083933898891814037984, 3.00719446299397611625270487578, 3.49281637379518849996451206796, 3.77751958887345520929614762693, 4.40117098225425295021103046427, 4.77871361540091166158362167198, 5.22441423566521426453473919341, 5.56302419554584969429267005843, 5.62516165285240581875011394205, 6.24778771839563916872662210302, 6.84645472681785116933849835324, 6.89850090292395935613614161426, 7.14596048291568799543740861323, 7.72323171442808617548991375098, 8.139653863252550983731007187116, 8.465228881280003051578712643536
