| L(s) = 1 | − 4·11-s − 6·13-s + 2·17-s − 8·19-s − 4·23-s − 6·29-s + 4·31-s + 2·37-s − 2·41-s − 12·43-s + 2·53-s + 4·59-s − 6·61-s − 4·67-s + 8·71-s + 6·73-s + 16·79-s − 4·83-s − 18·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 1.66·13-s + 0.485·17-s − 1.83·19-s − 0.834·23-s − 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.312·41-s − 1.82·43-s + 0.274·53-s + 0.520·59-s − 0.768·61-s − 0.488·67-s + 0.949·71-s + 0.702·73-s + 1.80·79-s − 0.439·83-s − 1.90·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.29989630081279, −12.88817925524109, −12.51792923310917, −12.13521241601464, −11.48801870895328, −11.12330411536522, −10.38363176547975, −10.06287920873662, −9.907524589797490, −9.111260478667703, −8.574022328210184, −8.079100984757359, −7.664832714625015, −7.275312997746144, −6.552714641811738, −6.181406340075739, −5.486648871962378, −4.982903186648741, −4.661615197743061, −3.971748798082462, −3.351122316033558, −2.668425604762867, −2.150683542552333, −1.809817461518069, −0.5521525087696206, 0,
0.5521525087696206, 1.809817461518069, 2.150683542552333, 2.668425604762867, 3.351122316033558, 3.971748798082462, 4.661615197743061, 4.982903186648741, 5.486648871962378, 6.181406340075739, 6.552714641811738, 7.275312997746144, 7.664832714625015, 8.079100984757359, 8.574022328210184, 9.111260478667703, 9.907524589797490, 10.06287920873662, 10.38363176547975, 11.12330411536522, 11.48801870895328, 12.13521241601464, 12.51792923310917, 12.88817925524109, 13.29989630081279
