| L(s) = 1 | + 1.67·3-s − 0.778·5-s + 3.24·7-s − 0.201·9-s − 5.19·11-s − 4.82·13-s − 1.30·15-s + 1.25·17-s − 19-s + 5.42·21-s + 8.02·23-s − 4.39·25-s − 5.35·27-s − 3.73·29-s + 8.22·31-s − 8.68·33-s − 2.52·35-s − 4.60·37-s − 8.07·39-s − 7.33·41-s − 7.17·43-s + 0.156·45-s + 2.69·47-s + 3.53·49-s + 2.09·51-s + 53-s + 4.04·55-s + ⋯ |
| L(s) = 1 | + 0.965·3-s − 0.348·5-s + 1.22·7-s − 0.0671·9-s − 1.56·11-s − 1.33·13-s − 0.336·15-s + 0.303·17-s − 0.229·19-s + 1.18·21-s + 1.67·23-s − 0.878·25-s − 1.03·27-s − 0.693·29-s + 1.47·31-s − 1.51·33-s − 0.426·35-s − 0.757·37-s − 1.29·39-s − 1.14·41-s − 1.09·43-s + 0.0233·45-s + 0.392·47-s + 0.504·49-s + 0.293·51-s + 0.137·53-s + 0.545·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 + 0.778T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 23 | \( 1 - 8.02T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 - 8.22T + 31T^{2} \) |
| 37 | \( 1 + 4.60T + 37T^{2} \) |
| 41 | \( 1 + 7.33T + 41T^{2} \) |
| 43 | \( 1 + 7.17T + 43T^{2} \) |
| 47 | \( 1 - 2.69T + 47T^{2} \) |
| 59 | \( 1 - 3.81T + 59T^{2} \) |
| 61 | \( 1 + 2.56T + 61T^{2} \) |
| 67 | \( 1 + 6.53T + 67T^{2} \) |
| 71 | \( 1 - 5.85T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 + 8.37T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 8.42T + 97T^{2} \) |
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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
−8.214338506071089551787408884016, −7.53737618245583612131567549625, −6.99692198084153288145720082127, −5.51385229987746975676808641586, −5.08450406471931880014120882152, −4.33322868140592553051449552393, −3.13140886953547584305951257600, −2.59890961805298718911393814692, −1.69689576641308101280967977533, 0,
1.69689576641308101280967977533, 2.59890961805298718911393814692, 3.13140886953547584305951257600, 4.33322868140592553051449552393, 5.08450406471931880014120882152, 5.51385229987746975676808641586, 6.99692198084153288145720082127, 7.53737618245583612131567549625, 8.214338506071089551787408884016
