| L(s) = 1 | + 1.82·5-s − 7-s − 4.82·11-s + 2.82·13-s + 0.171·17-s − 6.82·19-s + 4·23-s − 1.65·25-s + 2.82·29-s + 6.82·31-s − 1.82·35-s + 2.65·37-s + 3.82·41-s − 7.82·43-s − 8.65·47-s + 49-s − 2·53-s − 8.82·55-s − 0.656·59-s + 3.17·61-s + 5.17·65-s − 4·67-s − 1.65·71-s − 5.65·73-s + 4.82·77-s + 1.82·79-s + 5.34·83-s + ⋯ |
| L(s) = 1 | + 0.817·5-s − 0.377·7-s − 1.45·11-s + 0.784·13-s + 0.0416·17-s − 1.56·19-s + 0.834·23-s − 0.331·25-s + 0.525·29-s + 1.22·31-s − 0.309·35-s + 0.436·37-s + 0.597·41-s − 1.19·43-s − 1.26·47-s + 0.142·49-s − 0.274·53-s − 1.19·55-s − 0.0855·59-s + 0.406·61-s + 0.641·65-s − 0.488·67-s − 0.196·71-s − 0.662·73-s + 0.550·77-s + 0.205·79-s + 0.586·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.82T + 5T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 0.171T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 - 2.65T + 37T^{2} \) |
| 41 | \( 1 - 3.82T + 41T^{2} \) |
| 43 | \( 1 + 7.82T + 43T^{2} \) |
| 47 | \( 1 + 8.65T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 0.656T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 + 5.65T + 73T^{2} \) |
| 79 | \( 1 - 1.82T + 79T^{2} \) |
| 83 | \( 1 - 5.34T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.940251506125181964542625213198, −6.77298811774642900871713088120, −6.34995240345156980337986739286, −5.62417022239354461906383125881, −4.91841505741513666894215540823, −4.10583470214326168171154076130, −2.99228183304057857889543772363, −2.44097987761418239959035528206, −1.39680333501336359637658968782, 0,
1.39680333501336359637658968782, 2.44097987761418239959035528206, 2.99228183304057857889543772363, 4.10583470214326168171154076130, 4.91841505741513666894215540823, 5.62417022239354461906383125881, 6.34995240345156980337986739286, 6.77298811774642900871713088120, 7.940251506125181964542625213198
