L(s) = 1 | + 2·5-s + 7-s − 2·13-s − 6·17-s − 19-s − 25-s + 10·29-s − 4·31-s + 2·35-s + 2·37-s − 10·41-s + 4·43-s − 12·47-s + 49-s + 2·53-s + 4·59-s − 2·61-s − 4·65-s + 8·67-s − 6·73-s − 8·79-s + 4·83-s − 12·85-s − 10·89-s − 2·91-s − 2·95-s − 10·97-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.554·13-s − 1.45·17-s − 0.229·19-s − 1/5·25-s + 1.85·29-s − 0.718·31-s + 0.338·35-s + 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.274·53-s + 0.520·59-s − 0.256·61-s − 0.496·65-s + 0.977·67-s − 0.702·73-s − 0.900·79-s + 0.439·83-s − 1.30·85-s − 1.05·89-s − 0.209·91-s − 0.205·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.19899255097199131827096370217, −6.63067684020262337622495219531, −6.08036267745082153221102304655, −5.18270056391715484110268857661, −4.72327261341972902512535701683, −3.91423407912147102435589106022, −2.80451695918037226128151044714, −2.18041053621692159329207389435, −1.40295458051433947750080686556, 0,
1.40295458051433947750080686556, 2.18041053621692159329207389435, 2.80451695918037226128151044714, 3.91423407912147102435589106022, 4.72327261341972902512535701683, 5.18270056391715484110268857661, 6.08036267745082153221102304655, 6.63067684020262337622495219531, 7.19899255097199131827096370217