| L(s) = 1 | − 683·7-s − 4.03e3·13-s − 1.28e4·19-s + 1.56e4·25-s − 3.52e4·31-s + 3.02e3·37-s − 1.11e5·43-s + 3.48e5·49-s + 6.29e4·61-s + 5.85e5·67-s + 6.65e4·73-s − 9.37e5·79-s + 2.75e6·91-s − 1.55e6·97-s − 1.05e6·103-s − 2.17e6·109-s + ⋯ |
| L(s) = 1 | − 1.99·7-s − 1.83·13-s − 1.87·19-s + 25-s − 1.18·31-s + 0.0596·37-s − 1.40·43-s + 2.96·49-s + 0.277·61-s + 1.94·67-s + 0.171·73-s − 1.90·79-s + 3.65·91-s − 1.70·97-s − 0.968·103-s − 1.67·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4213833457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4213833457\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 7 | \( 1 + 683 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 + 4033 T + p^{6} T^{2} \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( 1 + 12851 T + p^{6} T^{2} \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 + 35282 T + p^{6} T^{2} \) |
| 37 | \( 1 - 3023 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 + 111386 T + p^{6} T^{2} \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 - 62999 T + p^{6} T^{2} \) |
| 67 | \( 1 - 585397 T + p^{6} T^{2} \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 - 66527 T + p^{6} T^{2} \) |
| 79 | \( 1 + 937691 T + p^{6} T^{2} \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( 1 + 1551817 T + p^{6} T^{2} \) |
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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
−10.00194167099947410154260534741, −9.459252464471088764111703418504, −8.470121051073500885066084656424, −7.05626257519246206190409858470, −6.65432049793436614425523558230, −5.48504771045733053324317447770, −4.26087019475839949124455517424, −3.12245878947342404099074283463, −2.23137566088707035244322440244, −0.28895755387969798083463640668,
0.28895755387969798083463640668, 2.23137566088707035244322440244, 3.12245878947342404099074283463, 4.26087019475839949124455517424, 5.48504771045733053324317447770, 6.65432049793436614425523558230, 7.05626257519246206190409858470, 8.470121051073500885066084656424, 9.459252464471088764111703418504, 10.00194167099947410154260534741
