| L(s) = 1 | − 2·3-s − 5-s − 4·7-s + 9-s − 11-s − 4·13-s + 2·15-s + 4·19-s + 8·21-s − 6·23-s + 25-s + 4·27-s + 6·29-s + 8·31-s + 2·33-s + 4·35-s − 2·37-s + 8·39-s − 6·41-s − 8·43-s − 45-s − 6·47-s + 9·49-s − 6·53-s + 55-s − 8·57-s + 12·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.516·15-s + 0.917·19-s + 1.74·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 1.43·31-s + 0.348·33-s + 0.676·35-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s − 0.824·53-s + 0.134·55-s − 1.05·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4468811347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4468811347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.50931985833896, −12.28177785933943, −12.04106231529824, −11.58025890306475, −11.08014809788472, −10.42853049106105, −9.972535342762937, −9.856733751900997, −9.384781604558488, −8.414719143428514, −8.248623891290479, −7.614075172238138, −6.868942930399780, −6.630505327079310, −6.365655973568613, −5.612039424789607, −5.173784132723349, −4.828086401717747, −4.133521269895492, −3.539797421471300, −2.923023396366990, −2.640888686801831, −1.688700457385729, −0.7607588112039889, −0.2772653736476592,
0.2772653736476592, 0.7607588112039889, 1.688700457385729, 2.640888686801831, 2.923023396366990, 3.539797421471300, 4.133521269895492, 4.828086401717747, 5.173784132723349, 5.612039424789607, 6.365655973568613, 6.630505327079310, 6.868942930399780, 7.614075172238138, 8.248623891290479, 8.414719143428514, 9.384781604558488, 9.856733751900997, 9.972535342762937, 10.42853049106105, 11.08014809788472, 11.58025890306475, 12.04106231529824, 12.28177785933943, 12.50931985833896
