| L(s) = 1 | + 0.823·2-s − 3-s − 1.32·4-s − 3.52·5-s − 0.823·6-s − 0.677·7-s − 2.73·8-s + 9-s − 2.90·10-s + 3.52·11-s + 1.32·12-s − 13-s − 0.558·14-s + 3.52·15-s + 0.391·16-s + 3.60·17-s + 0.823·18-s − 3.09·19-s + 4.66·20-s + 0.677·21-s + 2.90·22-s − 5.20·23-s + 2.73·24-s + 7.43·25-s − 0.823·26-s − 27-s + 0.896·28-s + ⋯ |
| L(s) = 1 | + 0.582·2-s − 0.577·3-s − 0.661·4-s − 1.57·5-s − 0.336·6-s − 0.256·7-s − 0.967·8-s + 0.333·9-s − 0.918·10-s + 1.06·11-s + 0.381·12-s − 0.277·13-s − 0.149·14-s + 0.910·15-s + 0.0979·16-s + 0.875·17-s + 0.194·18-s − 0.710·19-s + 1.04·20-s + 0.147·21-s + 0.618·22-s − 1.08·23-s + 0.558·24-s + 1.48·25-s − 0.161·26-s − 0.192·27-s + 0.169·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.823T + 2T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 7 | \( 1 + 0.677T + 7T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6.57T + 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 + 7.74T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 - 5.00T + 59T^{2} \) |
| 61 | \( 1 + 1.23T + 61T^{2} \) |
| 67 | \( 1 + 7.22T + 67T^{2} \) |
| 71 | \( 1 + 5.47T + 71T^{2} \) |
| 73 | \( 1 - 1.06T + 73T^{2} \) |
| 79 | \( 1 + 4.52T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 7.17T + 97T^{2} \) |
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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
−8.173682238481485440447702688685, −7.33287544991723082048931079563, −6.38725812802493083648188264186, −5.94122695388663792348054385337, −4.69622385924137080938294277755, −4.36995917094488078304525490184, −3.71111772258261195831199955403, −2.89116345399873289198033924561, −1.04562731709197022749480967718, 0,
1.04562731709197022749480967718, 2.89116345399873289198033924561, 3.71111772258261195831199955403, 4.36995917094488078304525490184, 4.69622385924137080938294277755, 5.94122695388663792348054385337, 6.38725812802493083648188264186, 7.33287544991723082048931079563, 8.173682238481485440447702688685
