| L(s) = 1 | − 3·4-s − 2·5-s + 5·16-s + 6·20-s − 4·23-s − 7·25-s − 4·31-s − 6·37-s − 4·47-s − 10·49-s − 18·53-s − 16·59-s − 3·64-s + 4·67-s − 24·71-s − 10·80-s + 18·89-s + 12·92-s − 26·97-s + 21·100-s + 16·103-s + 18·113-s + 8·115-s + 12·124-s + 26·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.894·5-s + 5/4·16-s + 1.34·20-s − 0.834·23-s − 7/5·25-s − 0.718·31-s − 0.986·37-s − 0.583·47-s − 1.42·49-s − 2.47·53-s − 2.08·59-s − 3/8·64-s + 0.488·67-s − 2.84·71-s − 1.11·80-s + 1.90·89-s + 1.25·92-s − 2.63·97-s + 2.09·100-s + 1.57·103-s + 1.69·113-s + 0.746·115-s + 1.07·124-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−7.69035551562908784013462192082, −7.43742239849917419501134588307, −6.74272981252089439582006678154, −6.02373904066748853888838701833, −5.97336865191641765106303088244, −5.21092257719357715807199072115, −4.68607726153727393582211790246, −4.50122243823954701224397495329, −3.96049656444482134108637352404, −3.33556810952068997685484956976, −3.26020657517957887232980598752, −2.02313919782203708817074154348, −1.45371564777531463821755560879, 0, 0,
1.45371564777531463821755560879, 2.02313919782203708817074154348, 3.26020657517957887232980598752, 3.33556810952068997685484956976, 3.96049656444482134108637352404, 4.50122243823954701224397495329, 4.68607726153727393582211790246, 5.21092257719357715807199072115, 5.97336865191641765106303088244, 6.02373904066748853888838701833, 6.74272981252089439582006678154, 7.43742239849917419501134588307, 7.69035551562908784013462192082
