| L(s) = 1 | + 1.48·3-s − 3.74·5-s + 4.87·7-s − 0.798·9-s + 4.72·11-s − 4.62·13-s − 5.55·15-s + 1.46·17-s − 7.83·19-s + 7.22·21-s − 4.86·23-s + 9.01·25-s − 5.63·27-s − 5.83·29-s + 3.71·31-s + 7.00·33-s − 18.2·35-s − 10.8·37-s − 6.86·39-s − 5.49·41-s + 4.74·43-s + 2.99·45-s + 12.1·47-s + 16.7·49-s + 2.16·51-s − 4.24·53-s − 17.6·55-s + ⋯ |
| L(s) = 1 | + 0.856·3-s − 1.67·5-s + 1.84·7-s − 0.266·9-s + 1.42·11-s − 1.28·13-s − 1.43·15-s + 0.354·17-s − 1.79·19-s + 1.57·21-s − 1.01·23-s + 1.80·25-s − 1.08·27-s − 1.08·29-s + 0.666·31-s + 1.21·33-s − 3.08·35-s − 1.77·37-s − 1.09·39-s − 0.857·41-s + 0.723·43-s + 0.445·45-s + 1.77·47-s + 2.39·49-s + 0.303·51-s − 0.582·53-s − 2.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.48T + 3T^{2} \) |
| 5 | \( 1 + 3.74T + 5T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 13 | \( 1 + 4.62T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 + 7.83T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + 5.83T + 29T^{2} \) |
| 31 | \( 1 - 3.71T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 5.49T + 41T^{2} \) |
| 43 | \( 1 - 4.74T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 4.78T + 59T^{2} \) |
| 61 | \( 1 + 3.97T + 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 + 2.93T + 71T^{2} \) |
| 73 | \( 1 + 5.86T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 2.77T + 83T^{2} \) |
| 89 | \( 1 + 1.93T + 89T^{2} \) |
| 97 | \( 1 + 9.38T + 97T^{2} \) |
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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
−8.141565731569080329983697731642, −7.53583274888709544894427155589, −7.03195026577583031039496211559, −5.78068044626985049795459981954, −4.70418693249244387157042873689, −4.18496396551127211333911415379, −3.64873066447829367116896763981, −2.41946668116275427057694255022, −1.60077071308458554953123399649, 0,
1.60077071308458554953123399649, 2.41946668116275427057694255022, 3.64873066447829367116896763981, 4.18496396551127211333911415379, 4.70418693249244387157042873689, 5.78068044626985049795459981954, 7.03195026577583031039496211559, 7.53583274888709544894427155589, 8.141565731569080329983697731642
