| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 4·11-s − 2·14-s + 16-s + 17-s + 4·19-s + 20-s − 4·22-s − 8·23-s + 25-s + 2·28-s − 8·29-s − 32-s − 34-s + 2·35-s + 2·37-s − 4·38-s − 40-s + 6·41-s + 2·43-s + 4·44-s + 8·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.20·11-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.377·28-s − 1.48·29-s − 0.176·32-s − 0.171·34-s + 0.338·35-s + 0.328·37-s − 0.648·38-s − 0.158·40-s + 0.937·41-s + 0.304·43-s + 0.603·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.03542149556723, −12.37318497073955, −12.06632332120524, −11.50372297194341, −11.35370404021783, −10.64909746399411, −10.26102056820089, −9.747748695068030, −9.171612856292170, −9.118536513893281, −8.443726100812260, −7.839232947011018, −7.418235094790177, −7.233031436889622, −6.269917803135188, −5.970381485077778, −5.684466543996042, −4.866451440754274, −4.327736120216801, −3.798618283977249, −3.253734037128273, −2.459616661466764, −1.929147508417787, −1.431110057134129, −0.9361079130432266, 0,
0.9361079130432266, 1.431110057134129, 1.929147508417787, 2.459616661466764, 3.253734037128273, 3.798618283977249, 4.327736120216801, 4.866451440754274, 5.684466543996042, 5.970381485077778, 6.269917803135188, 7.233031436889622, 7.418235094790177, 7.839232947011018, 8.443726100812260, 9.118536513893281, 9.171612856292170, 9.747748695068030, 10.26102056820089, 10.64909746399411, 11.35370404021783, 11.50372297194341, 12.06632332120524, 12.37318497073955, 13.03542149556723
