| L(s) = 1 | + 2-s − 1.86·3-s + 4-s − 3.25·5-s − 1.86·6-s − 2.35·7-s + 8-s + 0.463·9-s − 3.25·10-s + 4.08·11-s − 1.86·12-s − 5.56·13-s − 2.35·14-s + 6.06·15-s + 16-s − 1.83·17-s + 0.463·18-s + 3.58·19-s − 3.25·20-s + 4.38·21-s + 4.08·22-s + 4.99·23-s − 1.86·24-s + 5.61·25-s − 5.56·26-s + 4.72·27-s − 2.35·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.07·3-s + 0.5·4-s − 1.45·5-s − 0.759·6-s − 0.890·7-s + 0.353·8-s + 0.154·9-s − 1.03·10-s + 1.23·11-s − 0.537·12-s − 1.54·13-s − 0.629·14-s + 1.56·15-s + 0.250·16-s − 0.444·17-s + 0.109·18-s + 0.821·19-s − 0.728·20-s + 0.956·21-s + 0.870·22-s + 1.04·23-s − 0.379·24-s + 1.12·25-s − 1.09·26-s + 0.908·27-s − 0.445·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 1.86T + 3T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 - 4.08T + 11T^{2} \) |
| 13 | \( 1 + 5.56T + 13T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 - 4.99T + 23T^{2} \) |
| 29 | \( 1 - 0.324T + 29T^{2} \) |
| 37 | \( 1 - 9.87T + 37T^{2} \) |
| 41 | \( 1 + 0.175T + 41T^{2} \) |
| 43 | \( 1 - 2.08T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 7.85T + 53T^{2} \) |
| 59 | \( 1 + 3.88T + 59T^{2} \) |
| 61 | \( 1 + 9.78T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 6.00T + 71T^{2} \) |
| 73 | \( 1 + 1.96T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 + 4.54T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
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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
−7.31596681829320566777921837225, −7.00055389299314068303529162313, −6.28918881884548810045823668838, −5.57168631933391276887464987341, −4.63040332036369577908219839250, −4.32356551202884524499449184762, −3.31511555719722789241978833604, −2.71285688649857845314997606589, −1.00484666185742473691887430296, 0,
1.00484666185742473691887430296, 2.71285688649857845314997606589, 3.31511555719722789241978833604, 4.32356551202884524499449184762, 4.63040332036369577908219839250, 5.57168631933391276887464987341, 6.28918881884548810045823668838, 7.00055389299314068303529162313, 7.31596681829320566777921837225
