L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s − 4·11-s − 6·13-s − 2·15-s + 2·17-s − 4·19-s − 21-s − 8·23-s − 25-s − 27-s + 2·29-s + 4·33-s + 2·35-s + 10·37-s + 6·39-s − 6·41-s − 4·43-s + 2·45-s + 49-s − 2·51-s − 6·53-s − 8·55-s + 4·57-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.338·35-s + 1.64·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.563538518206812440434518911703, −8.186632228453794414425905803942, −7.68501515018855362022465153557, −6.60807660052202573549483918899, −5.76536999919480153066315644301, −5.11740288034969242991482089170, −4.30167637849672643861137972376, −2.66792336770788585023270622114, −1.88351369279783613545674963411, 0,
1.88351369279783613545674963411, 2.66792336770788585023270622114, 4.30167637849672643861137972376, 5.11740288034969242991482089170, 5.76536999919480153066315644301, 6.60807660052202573549483918899, 7.68501515018855362022465153557, 8.186632228453794414425905803942, 9.563538518206812440434518911703