L(s) = 1 | − 0.546·2-s − 3-s − 1.70·4-s + 0.546·6-s − 3.60·7-s + 2.02·8-s + 9-s + 2.27·11-s + 1.70·12-s − 0.323·13-s + 1.97·14-s + 2.29·16-s − 0.773·17-s − 0.546·18-s − 5.22·19-s + 3.60·21-s − 1.24·22-s − 0.497·23-s − 2.02·24-s + 0.176·26-s − 27-s + 6.12·28-s + 2.10·29-s + 0.935·31-s − 5.30·32-s − 2.27·33-s + 0.423·34-s + ⋯ |
L(s) = 1 | − 0.386·2-s − 0.577·3-s − 0.850·4-s + 0.223·6-s − 1.36·7-s + 0.715·8-s + 0.333·9-s + 0.687·11-s + 0.490·12-s − 0.0896·13-s + 0.526·14-s + 0.573·16-s − 0.187·17-s − 0.128·18-s − 1.19·19-s + 0.786·21-s − 0.265·22-s − 0.103·23-s − 0.413·24-s + 0.0346·26-s − 0.192·27-s + 1.15·28-s + 0.390·29-s + 0.168·31-s − 0.937·32-s − 0.396·33-s + 0.0725·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 67 | \( 1 + T \) |
good | 2 | \( 1 + 0.546T + 2T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 + 0.323T + 13T^{2} \) |
| 17 | \( 1 + 0.773T + 17T^{2} \) |
| 19 | \( 1 + 5.22T + 19T^{2} \) |
| 23 | \( 1 + 0.497T + 23T^{2} \) |
| 29 | \( 1 - 2.10T + 29T^{2} \) |
| 31 | \( 1 - 0.935T + 31T^{2} \) |
| 37 | \( 1 - 0.302T + 37T^{2} \) |
| 41 | \( 1 - 7.63T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 1.57T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 1.83T + 61T^{2} \) |
| 71 | \( 1 - 4.03T + 71T^{2} \) |
| 73 | \( 1 - 3.28T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 3.86T + 89T^{2} \) |
| 97 | \( 1 - 5.57T + 97T^{2} \) |
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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
−7.981729017928230673476975886560, −7.00529032613865177687275467504, −6.48525033373116018305636108966, −5.81447015110342805865820397249, −4.90781449276700175256199045702, −4.10395130821112177263404004881, −3.54683617800367158940418081705, −2.30022585241743985076469082320, −0.959126856659425587800279109177, 0,
0.959126856659425587800279109177, 2.30022585241743985076469082320, 3.54683617800367158940418081705, 4.10395130821112177263404004881, 4.90781449276700175256199045702, 5.81447015110342805865820397249, 6.48525033373116018305636108966, 7.00529032613865177687275467504, 7.981729017928230673476975886560