| L(s) = 1 | − 3-s − 3.46·5-s + 7-s + 9-s − 5.46·11-s + 2·13-s + 3.46·15-s + 7.46·17-s + 6.92·19-s − 21-s − 5.46·23-s + 6.99·25-s − 27-s + 8.92·29-s − 2.92·31-s + 5.46·33-s − 3.46·35-s − 2·37-s − 2·39-s + 4.53·41-s + 8·43-s − 3.46·45-s + 2.92·47-s + 49-s − 7.46·51-s − 2·53-s + 18.9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.54·5-s + 0.377·7-s + 0.333·9-s − 1.64·11-s + 0.554·13-s + 0.894·15-s + 1.81·17-s + 1.58·19-s − 0.218·21-s − 1.13·23-s + 1.39·25-s − 0.192·27-s + 1.65·29-s − 0.525·31-s + 0.951·33-s − 0.585·35-s − 0.328·37-s − 0.320·39-s + 0.708·41-s + 1.21·43-s − 0.516·45-s + 0.427·47-s + 0.142·49-s − 1.04·51-s − 0.274·53-s + 2.55·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8895287965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8895287965\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 + 2.92T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
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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
−10.57631482453243474378183203119, −9.938297427397215762577100402412, −8.446193395623413813551306595805, −7.73494323047549860655689075847, −7.37688967055536977753915925152, −5.79129763650381679010988933570, −5.09065120399941141972797763389, −3.97919901131423616113506293093, −2.97397759578997983391736542258, −0.836291699434975783758060568222,
0.836291699434975783758060568222, 2.97397759578997983391736542258, 3.97919901131423616113506293093, 5.09065120399941141972797763389, 5.79129763650381679010988933570, 7.37688967055536977753915925152, 7.73494323047549860655689075847, 8.446193395623413813551306595805, 9.938297427397215762577100402412, 10.57631482453243474378183203119
