| L(s) = 1 | − 2·5-s − 4·7-s − 11-s − 6·17-s + 8·19-s − 25-s + 6·29-s + 8·35-s − 6·37-s − 10·41-s − 8·43-s + 9·49-s − 6·53-s + 2·55-s + 4·59-s − 2·61-s + 12·67-s − 8·71-s − 2·73-s + 4·77-s − 4·79-s − 12·83-s + 12·85-s − 6·89-s − 16·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 0.301·11-s − 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.35·35-s − 0.986·37-s − 1.56·41-s − 1.21·43-s + 9/7·49-s − 0.824·53-s + 0.269·55-s + 0.520·59-s − 0.256·61-s + 1.46·67-s − 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.450·79-s − 1.31·83-s + 1.30·85-s − 0.635·89-s − 1.64·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.76283953008335, −13.43362201033074, −13.06406839730876, −12.39720799840000, −12.05808243705163, −11.56657253874644, −11.20819286155248, −10.45738958173385, −10.03548961338721, −9.651225369566529, −9.126333341645593, −8.473962427861108, −8.211428090769533, −7.439326256690090, −6.914310048902435, −6.730135613345590, −6.090840929730868, −5.345893379909796, −4.955587147863468, −4.199440303704932, −3.713086184944207, −3.074397319512817, −2.897984905196800, −1.928981263918752, −1.095381795090445, 0, 0,
1.095381795090445, 1.928981263918752, 2.897984905196800, 3.074397319512817, 3.713086184944207, 4.199440303704932, 4.955587147863468, 5.345893379909796, 6.090840929730868, 6.730135613345590, 6.914310048902435, 7.439326256690090, 8.211428090769533, 8.473962427861108, 9.126333341645593, 9.651225369566529, 10.03548961338721, 10.45738958173385, 11.20819286155248, 11.56657253874644, 12.05808243705163, 12.39720799840000, 13.06406839730876, 13.43362201033074, 13.76283953008335
