| L(s) = 1 | + 2·3-s − 3·5-s − 7-s + 3·9-s − 3·11-s − 4·13-s − 6·15-s − 3·17-s + 2·19-s − 2·21-s + 6·23-s + 5·25-s + 4·27-s + 6·29-s + 2·31-s − 6·33-s + 3·35-s + 2·37-s − 8·39-s + 43-s − 9·45-s + 21·47-s − 5·49-s − 6·51-s − 6·53-s + 9·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s − 0.377·7-s + 9-s − 0.904·11-s − 1.10·13-s − 1.54·15-s − 0.727·17-s + 0.458·19-s − 0.436·21-s + 1.25·23-s + 25-s + 0.769·27-s + 1.11·29-s + 0.359·31-s − 1.04·33-s + 0.507·35-s + 0.328·37-s − 1.28·39-s + 0.152·43-s − 1.34·45-s + 3.06·47-s − 5/7·49-s − 0.840·51-s − 0.824·53-s + 1.21·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13307904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13307904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.879411388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.879411388\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - T + 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 21 T + 196 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 144 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 144 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.937481475739076679677312442709, −8.248791894023842577276140184509, −7.931748146215572324967528091695, −7.65707158045090987352324061203, −7.39187962057711641285735973685, −7.09294100910721360577997653838, −6.60423406114954527848891637835, −6.36332181631333601808371888601, −5.64327405679876956912533386039, −5.14777342748686223027653347860, −4.71803539183838485494207090690, −4.63633768345285786441848293949, −3.97406039851705604610980165888, −3.67414152563806092250584762789, −3.01509421395640737896983208314, −2.99702120739305398395877308874, −2.21978354355464616441804742205, −2.18928595452425745194742879847, −0.875970446265646367019217024374, −0.60810774039772704370668758240,
0.60810774039772704370668758240, 0.875970446265646367019217024374, 2.18928595452425745194742879847, 2.21978354355464616441804742205, 2.99702120739305398395877308874, 3.01509421395640737896983208314, 3.67414152563806092250584762789, 3.97406039851705604610980165888, 4.63633768345285786441848293949, 4.71803539183838485494207090690, 5.14777342748686223027653347860, 5.64327405679876956912533386039, 6.36332181631333601808371888601, 6.60423406114954527848891637835, 7.09294100910721360577997653838, 7.39187962057711641285735973685, 7.65707158045090987352324061203, 7.931748146215572324967528091695, 8.248791894023842577276140184509, 8.937481475739076679677312442709
