| L(s) = 1 | + 0.563·2-s + 3-s − 1.68·4-s + 0.563·6-s − 1.09·7-s − 2.07·8-s + 9-s + 0.827·11-s − 1.68·12-s − 2.66·13-s − 0.617·14-s + 2.19·16-s + 4.16·17-s + 0.563·18-s − 5.48·19-s − 1.09·21-s + 0.466·22-s + 5.61·23-s − 2.07·24-s − 1.50·26-s + 27-s + 1.84·28-s + 9.79·29-s − 9.33·31-s + 5.38·32-s + 0.827·33-s + 2.34·34-s + ⋯ |
| L(s) = 1 | + 0.398·2-s + 0.577·3-s − 0.841·4-s + 0.230·6-s − 0.414·7-s − 0.733·8-s + 0.333·9-s + 0.249·11-s − 0.485·12-s − 0.738·13-s − 0.164·14-s + 0.549·16-s + 1.01·17-s + 0.132·18-s − 1.25·19-s − 0.239·21-s + 0.0993·22-s + 1.17·23-s − 0.423·24-s − 0.294·26-s + 0.192·27-s + 0.348·28-s + 1.81·29-s − 1.67·31-s + 0.952·32-s + 0.144·33-s + 0.402·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.563T + 2T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 11 | \( 1 - 0.827T + 11T^{2} \) |
| 13 | \( 1 + 2.66T + 13T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 - 5.61T + 23T^{2} \) |
| 29 | \( 1 - 9.79T + 29T^{2} \) |
| 31 | \( 1 + 9.33T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 2.92T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 9.17T + 59T^{2} \) |
| 61 | \( 1 + 4.17T + 61T^{2} \) |
| 67 | \( 1 + 7.79T + 67T^{2} \) |
| 71 | \( 1 + 8.28T + 71T^{2} \) |
| 73 | \( 1 + 4.51T + 73T^{2} \) |
| 79 | \( 1 - 3.96T + 79T^{2} \) |
| 83 | \( 1 + 9.53T + 83T^{2} \) |
| 89 | \( 1 - 4.86T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.324868228835608550919148864536, −7.51222990083330237723696173659, −6.70676520731895314579199378448, −5.87308969341400344964691972040, −4.95515250519028717074905790212, −4.39681957504618010800187037656, −3.39220916636590800436625533907, −2.89163403325775064818013564253, −1.50176951268586183272098349484, 0,
1.50176951268586183272098349484, 2.89163403325775064818013564253, 3.39220916636590800436625533907, 4.39681957504618010800187037656, 4.95515250519028717074905790212, 5.87308969341400344964691972040, 6.70676520731895314579199378448, 7.51222990083330237723696173659, 8.324868228835608550919148864536
