| L(s) = 1 | − 2-s + 4-s + 2.73·7-s − 8-s − 0.267·11-s − 0.732·13-s − 2.73·14-s + 16-s + 4.19·17-s − 19-s + 0.267·22-s − 7.92·23-s + 0.732·26-s + 2.73·28-s − 1.73·29-s + 4.46·31-s − 32-s − 4.19·34-s + 2·37-s + 38-s − 10.9·41-s − 2.19·43-s − 0.267·44-s + 7.92·46-s − 3.46·47-s + 0.464·49-s − 0.732·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.03·7-s − 0.353·8-s − 0.0807·11-s − 0.203·13-s − 0.730·14-s + 0.250·16-s + 1.01·17-s − 0.229·19-s + 0.0571·22-s − 1.65·23-s + 0.143·26-s + 0.516·28-s − 0.321·29-s + 0.801·31-s − 0.176·32-s − 0.719·34-s + 0.328·37-s + 0.162·38-s − 1.70·41-s − 0.334·43-s − 0.0403·44-s + 1.16·46-s − 0.505·47-s + 0.0663·49-s − 0.101·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 0.267T + 11T^{2} \) |
| 13 | \( 1 + 0.732T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 23 | \( 1 + 7.92T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 2.19T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 1.73T + 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 6.66T + 61T^{2} \) |
| 67 | \( 1 - 3.73T + 67T^{2} \) |
| 71 | \( 1 + 1.80T + 71T^{2} \) |
| 73 | \( 1 + 4.46T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 0.267T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 9.12T + 97T^{2} \) |
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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
−7.74125994588042306531232476338, −6.87176852707606962972417218936, −6.12538314503489766221221233957, −5.41497128270976921360655609535, −4.69346353739913181874572175716, −3.84457553767279559369535055902, −2.92328063462744528671562444541, −1.95314250417126341780329729659, −1.32259531930289380905985966900, 0,
1.32259531930289380905985966900, 1.95314250417126341780329729659, 2.92328063462744528671562444541, 3.84457553767279559369535055902, 4.69346353739913181874572175716, 5.41497128270976921360655609535, 6.12538314503489766221221233957, 6.87176852707606962972417218936, 7.74125994588042306531232476338
