| L(s) = 1 | + 1.66·2-s − 3.15·3-s + 0.760·4-s + 0.0956·5-s − 5.24·6-s + 3.17·7-s − 2.05·8-s + 6.95·9-s + 0.158·10-s − 3.11·11-s − 2.40·12-s − 4.72·13-s + 5.27·14-s − 0.301·15-s − 4.94·16-s + 17-s + 11.5·18-s + 2.56·19-s + 0.0727·20-s − 10.0·21-s − 5.18·22-s − 1.94·23-s + 6.49·24-s − 4.99·25-s − 7.84·26-s − 12.4·27-s + 2.41·28-s + ⋯ |
| L(s) = 1 | + 1.17·2-s − 1.82·3-s + 0.380·4-s + 0.0427·5-s − 2.14·6-s + 1.20·7-s − 0.727·8-s + 2.31·9-s + 0.0502·10-s − 0.940·11-s − 0.693·12-s − 1.30·13-s + 1.41·14-s − 0.0779·15-s − 1.23·16-s + 0.242·17-s + 2.72·18-s + 0.587·19-s + 0.0162·20-s − 2.18·21-s − 1.10·22-s − 0.405·23-s + 1.32·24-s − 0.998·25-s − 1.53·26-s − 2.40·27-s + 0.456·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.66T + 2T^{2} \) |
| 3 | \( 1 + 3.15T + 3T^{2} \) |
| 5 | \( 1 - 0.0956T + 5T^{2} \) |
| 7 | \( 1 - 3.17T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 + 4.72T + 13T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 + 1.94T + 23T^{2} \) |
| 29 | \( 1 + 9.57T + 29T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 0.0367T + 41T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 - 9.01T + 53T^{2} \) |
| 59 | \( 1 + 0.151T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 8.28T + 67T^{2} \) |
| 71 | \( 1 - 9.37T + 71T^{2} \) |
| 73 | \( 1 - 0.228T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−10.34198751100860353387238972261, −9.395878116986276924169461732426, −7.79510324028773549840047029415, −7.11669973664803332310045326623, −5.85552069170952997473946586634, −5.25755184216807165941788283279, −4.92039393044154557176514529075, −3.85261992760751637288496706523, −2.01647907485953133685511696937, 0,
2.01647907485953133685511696937, 3.85261992760751637288496706523, 4.92039393044154557176514529075, 5.25755184216807165941788283279, 5.85552069170952997473946586634, 7.11669973664803332310045326623, 7.79510324028773549840047029415, 9.395878116986276924169461732426, 10.34198751100860353387238972261
