| L(s) = 1 | − 5-s + 2·11-s + 13-s + 17-s + 4·19-s + 2·23-s + 25-s + 4·31-s + 6·37-s + 8·41-s − 4·43-s − 8·47-s − 7·49-s + 2·53-s − 2·55-s − 65-s + 8·67-s + 12·71-s − 16·73-s − 10·79-s − 85-s + 10·89-s − 4·95-s − 16·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s + 0.277·13-s + 0.242·17-s + 0.917·19-s + 0.417·23-s + 1/5·25-s + 0.718·31-s + 0.986·37-s + 1.24·41-s − 0.609·43-s − 1.16·47-s − 49-s + 0.274·53-s − 0.269·55-s − 0.124·65-s + 0.977·67-s + 1.42·71-s − 1.87·73-s − 1.12·79-s − 0.108·85-s + 1.05·89-s − 0.410·95-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.60935415322878, −12.92375833124203, −12.63375447210814, −12.05723931079420, −11.44657450923901, −11.36669678339960, −10.80258299591119, −10.02115331584070, −9.744407122448855, −9.257382204222291, −8.674262421623760, −8.158517255162547, −7.780806648921213, −7.197221529231906, −6.676851302256775, −6.241828236194758, −5.607712211522078, −5.090558927402013, −4.460357069801074, −4.043409760373768, −3.333701633626797, −2.971886097151711, −2.237177436754939, −1.316047534289866, −0.9814162675839779, 0,
0.9814162675839779, 1.316047534289866, 2.237177436754939, 2.971886097151711, 3.333701633626797, 4.043409760373768, 4.460357069801074, 5.090558927402013, 5.607712211522078, 6.241828236194758, 6.676851302256775, 7.197221529231906, 7.780806648921213, 8.158517255162547, 8.674262421623760, 9.257382204222291, 9.744407122448855, 10.02115331584070, 10.80258299591119, 11.36669678339960, 11.44657450923901, 12.05723931079420, 12.63375447210814, 12.92375833124203, 13.60935415322878
