L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s − 13-s − 17-s − 6·19-s + 2·21-s + 8·23-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 2·37-s + 39-s − 2·41-s − 10·43-s − 4·47-s − 3·49-s + 51-s + 2·53-s + 6·57-s + 14·59-s − 10·61-s − 2·63-s + 12·67-s − 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.242·17-s − 1.37·19-s + 0.436·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.160·39-s − 0.312·41-s − 1.52·43-s − 0.583·47-s − 3/7·49-s + 0.140·51-s + 0.274·53-s + 0.794·57-s + 1.82·59-s − 1.28·61-s − 0.251·63-s + 1.46·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.89149884368283, −12.73272589220000, −12.17149913684048, −11.61302019701396, −11.12094905966910, −10.78537319091946, −10.11010442345842, −10.03064573481622, −9.509089925027143, −8.667681573874535, −8.421727285822796, −8.010887285066569, −7.165348617853432, −6.864803329757206, −6.434618225491900, −6.027793693051681, −5.262903143244612, −4.906026315782800, −4.558235775635974, −3.830915656333933, −3.128660352961745, −2.710184664392117, −2.224519779987872, −1.339082700700702, −0.6119253532084352, 0,
0.6119253532084352, 1.339082700700702, 2.224519779987872, 2.710184664392117, 3.128660352961745, 3.830915656333933, 4.558235775635974, 4.906026315782800, 5.262903143244612, 6.027793693051681, 6.434618225491900, 6.864803329757206, 7.165348617853432, 8.010887285066569, 8.421727285822796, 8.667681573874535, 9.509089925027143, 10.03064573481622, 10.11010442345842, 10.78537319091946, 11.12094905966910, 11.61302019701396, 12.17149913684048, 12.73272589220000, 12.89149884368283