| L(s) = 1 | − 8·5-s + 9·9-s − 24·13-s + 16·17-s + 39·25-s − 42·29-s + 70·37-s + 80·41-s − 72·45-s + 90·53-s + 120·61-s + 192·65-s + 96·73-s + 81·81-s − 128·85-s − 160·89-s − 144·97-s − 40·101-s + 182·109-s − 30·113-s − 216·117-s + ⋯ |
| L(s) = 1 | − 8/5·5-s + 9-s − 1.84·13-s + 0.941·17-s + 1.55·25-s − 1.44·29-s + 1.89·37-s + 1.95·41-s − 8/5·45-s + 1.69·53-s + 1.96·61-s + 2.95·65-s + 1.31·73-s + 81-s − 1.50·85-s − 1.79·89-s − 1.48·97-s − 0.396·101-s + 1.66·109-s − 0.265·113-s − 1.84·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.189337803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.189337803\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( 1 + 8 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 24 T + p^{2} T^{2} \) |
| 17 | \( 1 - 16 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 42 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 70 T + p^{2} T^{2} \) |
| 41 | \( 1 - 80 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 90 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 120 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 96 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 160 T + p^{2} T^{2} \) |
| 97 | \( 1 + 144 T + p^{2} T^{2} \) |
| 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
−9.984585550284548620294217065085, −9.408881604044397990361414200344, −8.072805259291288642002801669653, −7.49971647495760473194397247034, −7.04361025266016698311353104785, −5.49904266786644395095889145941, −4.44187173240799979717417199936, −3.82857336544422424437801668356, −2.50767016193967557090823352671, −0.70796620604602728574649380204,
0.70796620604602728574649380204, 2.50767016193967557090823352671, 3.82857336544422424437801668356, 4.44187173240799979717417199936, 5.49904266786644395095889145941, 7.04361025266016698311353104785, 7.49971647495760473194397247034, 8.072805259291288642002801669653, 9.408881604044397990361414200344, 9.984585550284548620294217065085
