L(s) = 1 | − 1.49·2-s + 3.12·3-s + 0.226·4-s − 4.66·6-s + 4.66·7-s + 2.64·8-s + 6.77·9-s − 5.92·11-s + 0.707·12-s − 4.29·13-s − 6.95·14-s − 4.40·16-s − 5.72·17-s − 10.1·18-s − 1.23·19-s + 14.5·21-s + 8.83·22-s − 3.63·23-s + 8.27·24-s + 6.40·26-s + 11.8·27-s + 1.05·28-s + 1.81·29-s − 1.11·31-s + 1.27·32-s − 18.5·33-s + 8.54·34-s + ⋯ |
L(s) = 1 | − 1.05·2-s + 1.80·3-s + 0.113·4-s − 1.90·6-s + 1.76·7-s + 0.935·8-s + 2.25·9-s − 1.78·11-s + 0.204·12-s − 1.19·13-s − 1.85·14-s − 1.10·16-s − 1.38·17-s − 2.38·18-s − 0.282·19-s + 3.17·21-s + 1.88·22-s − 0.757·23-s + 1.68·24-s + 1.25·26-s + 2.27·27-s + 0.199·28-s + 0.336·29-s − 0.200·31-s + 0.225·32-s − 3.22·33-s + 1.46·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 1.49T + 2T^{2} \) |
| 3 | \( 1 - 3.12T + 3T^{2} \) |
| 7 | \( 1 - 4.66T + 7T^{2} \) |
| 11 | \( 1 + 5.92T + 11T^{2} \) |
| 13 | \( 1 + 4.29T + 13T^{2} \) |
| 17 | \( 1 + 5.72T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 + 3.63T + 23T^{2} \) |
| 29 | \( 1 - 1.81T + 29T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 + 7.32T + 37T^{2} \) |
| 41 | \( 1 + 6.45T + 41T^{2} \) |
| 43 | \( 1 + 4.57T + 43T^{2} \) |
| 47 | \( 1 + 7.71T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 - 4.76T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 3.96T + 71T^{2} \) |
| 73 | \( 1 - 8.92T + 73T^{2} \) |
| 79 | \( 1 - 1.77T + 79T^{2} \) |
| 83 | \( 1 + 9.93T + 83T^{2} \) |
| 89 | \( 1 + 7.03T + 89T^{2} \) |
| 97 | \( 1 - 5.03T + 97T^{2} \) |
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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
−8.016042028848617839965199968453, −7.46028137922998920686612562741, −6.97122015804690543738151722972, −5.18645431469153560354813294931, −4.75443564157828115168863241380, −4.10259351629519217822075822179, −2.80970906459896170130032202826, −2.08023430522784944238590137448, −1.71727628992969244767994518416, 0,
1.71727628992969244767994518416, 2.08023430522784944238590137448, 2.80970906459896170130032202826, 4.10259351629519217822075822179, 4.75443564157828115168863241380, 5.18645431469153560354813294931, 6.97122015804690543738151722972, 7.46028137922998920686612562741, 8.016042028848617839965199968453