| L(s) = 1 | − 2·5-s − 9-s + 4·11-s + 12·19-s − 25-s + 12·29-s + 20·31-s + 12·41-s + 2·45-s − 49-s − 8·55-s − 12·61-s + 12·71-s − 8·79-s + 81-s + 12·89-s − 24·95-s − 4·99-s − 20·101-s − 20·109-s − 10·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1/3·9-s + 1.20·11-s + 2.75·19-s − 1/5·25-s + 2.22·29-s + 3.59·31-s + 1.87·41-s + 0.298·45-s − 1/7·49-s − 1.07·55-s − 1.53·61-s + 1.42·71-s − 0.900·79-s + 1/9·81-s + 1.27·89-s − 2.46·95-s − 0.402·99-s − 1.99·101-s − 1.91·109-s − 0.909·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.227252818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.227252818\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.24622376850162292838146886493, −10.02695330967953646866738060766, −9.418011676044524388548164316249, −9.357583811269004123427876476300, −8.663249985612152757254746691212, −8.152879485491472851596332374530, −7.943842594689651466213394460428, −7.58537639148432645329750212349, −6.78580328359841470632783419403, −6.73412513690147874693435498117, −6.04024853233344267166078629801, −5.71681709487707030678622672176, −4.78787157450412168484313680364, −4.71959877830515934641097957805, −4.07404136542327940724636991805, −3.51748428969480125427180431368, −2.87066147237433745763767703463, −2.65807631367547533223490342958, −1.14211722010438502308014106984, −0.968338796837998335226964977087,
0.968338796837998335226964977087, 1.14211722010438502308014106984, 2.65807631367547533223490342958, 2.87066147237433745763767703463, 3.51748428969480125427180431368, 4.07404136542327940724636991805, 4.71959877830515934641097957805, 4.78787157450412168484313680364, 5.71681709487707030678622672176, 6.04024853233344267166078629801, 6.73412513690147874693435498117, 6.78580328359841470632783419403, 7.58537639148432645329750212349, 7.943842594689651466213394460428, 8.152879485491472851596332374530, 8.663249985612152757254746691212, 9.357583811269004123427876476300, 9.418011676044524388548164316249, 10.02695330967953646866738060766, 10.24622376850162292838146886493
