| L(s) = 1 | − 3.46·7-s + 6.92·13-s − 8·19-s − 5·25-s + 10.3·31-s − 6.92·37-s − 8·43-s + 4.99·49-s − 6.92·61-s − 16·67-s + 10·73-s − 17.3·79-s − 23.9·91-s − 14·97-s + 3.46·103-s − 20.7·109-s + ⋯ |
| L(s) = 1 | − 1.30·7-s + 1.92·13-s − 1.83·19-s − 25-s + 1.86·31-s − 1.13·37-s − 1.21·43-s + 0.714·49-s − 0.887·61-s − 1.95·67-s + 1.17·73-s − 1.94·79-s − 2.51·91-s − 1.42·97-s + 0.341·103-s − 1.99·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.92T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 + 16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.593793309595105348198669578322, −8.077355162777731882187280732349, −6.73600296491697091426064441156, −6.38267999972449632798926337798, −5.73545497379293359491278844086, −4.38600891961246791504926911244, −3.69163750783879905053542927413, −2.84939932663835046383693403036, −1.54730251588345814331336383544, 0,
1.54730251588345814331336383544, 2.84939932663835046383693403036, 3.69163750783879905053542927413, 4.38600891961246791504926911244, 5.73545497379293359491278844086, 6.38267999972449632798926337798, 6.73600296491697091426064441156, 8.077355162777731882187280732349, 8.593793309595105348198669578322
