| L(s) = 1 | − 14·7-s + 36·23-s − 50·25-s + 147·49-s − 228·71-s − 188·79-s − 444·113-s − 206·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 504·161-s + 163-s + 167-s − 338·169-s + 173-s + 700·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2·7-s + 1.56·23-s − 2·25-s + 3·49-s − 3.21·71-s − 2.37·79-s − 3.92·113-s − 1.70·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s − 3.13·161-s + 0.00613·163-s + 0.00598·167-s − 2·169-s + 0.00578·173-s + 4·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01214597878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01214597878\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )( 1 + 6 T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )( 1 + 54 T + p^{2} T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )( 1 + 38 T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )( 1 + 58 T + p^{2} T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )( 1 + 6 T + p^{2} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 118 T + p^{2} T^{2} )( 1 + 118 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 114 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−9.134855363953494500969859065174, −8.846452863248833817514887084867, −8.618954741212006329436405442112, −7.75992266973846955055913343283, −7.67880669113241881892674705465, −7.12203772849200735743861350529, −6.81157135663284652659516753853, −6.43119356081002586495334288577, −6.03031052514452726502199201620, −5.54571556520183133753340853248, −5.44455417532927665519004380748, −4.55285009329449832963571490906, −4.22725012430522133297992461543, −3.70799097974886596535218645797, −3.34446239092628467624939635321, −2.69423066407047643920099584640, −2.64365682634996422345874010532, −1.65034391567041123838731432760, −1.07069985503597747408623604801, −0.03035911177150904451252451532,
0.03035911177150904451252451532, 1.07069985503597747408623604801, 1.65034391567041123838731432760, 2.64365682634996422345874010532, 2.69423066407047643920099584640, 3.34446239092628467624939635321, 3.70799097974886596535218645797, 4.22725012430522133297992461543, 4.55285009329449832963571490906, 5.44455417532927665519004380748, 5.54571556520183133753340853248, 6.03031052514452726502199201620, 6.43119356081002586495334288577, 6.81157135663284652659516753853, 7.12203772849200735743861350529, 7.67880669113241881892674705465, 7.75992266973846955055913343283, 8.618954741212006329436405442112, 8.846452863248833817514887084867, 9.134855363953494500969859065174
