| L(s) = 1 | − 5-s + 6·17-s − 4·19-s + 25-s + 2·29-s + 2·37-s − 2·41-s − 4·43-s + 4·47-s − 7·49-s + 10·53-s + 8·59-s − 2·61-s + 4·67-s − 12·71-s + 6·73-s − 16·83-s − 6·85-s − 10·89-s + 4·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 49-s + 1.37·53-s + 1.04·59-s − 0.256·61-s + 0.488·67-s − 1.42·71-s + 0.702·73-s − 1.75·83-s − 0.650·85-s − 1.05·89-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.62653301522286, −14.13069571574099, −13.51851560717663, −12.98600004425497, −12.48868052295855, −12.05276296482323, −11.54044367732055, −11.06968900957357, −10.36315149274344, −10.05362229402391, −9.513388448904866, −8.695224999738117, −8.414237516919908, −7.828819709240526, −7.276443369981650, −6.754477748442332, −6.121691110719434, −5.515535215177466, −5.014961039267487, −4.237643016474681, −3.822313632434255, −3.093905360495723, −2.536049927611069, −1.636274357960329, −0.9318376170380364, 0,
0.9318376170380364, 1.636274357960329, 2.536049927611069, 3.093905360495723, 3.822313632434255, 4.237643016474681, 5.014961039267487, 5.515535215177466, 6.121691110719434, 6.754477748442332, 7.276443369981650, 7.828819709240526, 8.414237516919908, 8.695224999738117, 9.513388448904866, 10.05362229402391, 10.36315149274344, 11.06968900957357, 11.54044367732055, 12.05276296482323, 12.48868052295855, 12.98600004425497, 13.51851560717663, 14.13069571574099, 14.62653301522286
