L(s) = 1 | + 0.419·2-s + 3-s − 1.82·4-s + 0.277·5-s + 0.419·6-s + 7-s − 1.60·8-s + 9-s + 0.116·10-s + 4.81·11-s − 1.82·12-s − 0.382·13-s + 0.419·14-s + 0.277·15-s + 2.97·16-s − 2.75·17-s + 0.419·18-s − 3.49·19-s − 0.505·20-s + 21-s + 2.02·22-s − 1.28·23-s − 1.60·24-s − 4.92·25-s − 0.160·26-s + 27-s − 1.82·28-s + ⋯ |
L(s) = 1 | + 0.296·2-s + 0.577·3-s − 0.912·4-s + 0.123·5-s + 0.171·6-s + 0.377·7-s − 0.566·8-s + 0.333·9-s + 0.0367·10-s + 1.45·11-s − 0.526·12-s − 0.106·13-s + 0.112·14-s + 0.0715·15-s + 0.744·16-s − 0.667·17-s + 0.0988·18-s − 0.800·19-s − 0.112·20-s + 0.218·21-s + 0.430·22-s − 0.267·23-s − 0.327·24-s − 0.984·25-s − 0.0314·26-s + 0.192·27-s − 0.344·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 - 0.419T + 2T^{2} \) |
| 5 | \( 1 - 0.277T + 5T^{2} \) |
| 11 | \( 1 - 4.81T + 11T^{2} \) |
| 13 | \( 1 + 0.382T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 19 | \( 1 + 3.49T + 19T^{2} \) |
| 23 | \( 1 + 1.28T + 23T^{2} \) |
| 29 | \( 1 + 9.01T + 29T^{2} \) |
| 31 | \( 1 + 0.526T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 + 5.83T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 1.83T + 53T^{2} \) |
| 59 | \( 1 - 9.44T + 59T^{2} \) |
| 61 | \( 1 + 1.63T + 61T^{2} \) |
| 67 | \( 1 + 6.87T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 5.24T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 - 2.81T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−7.62471887998258366822358681114, −6.72454424125334333466695346352, −6.07799242721001796498360652148, −5.32693634898356568211718896894, −4.37446907339856385467029983678, −4.04589116505934935133784017202, −3.35115029942198380223595347013, −2.17354860040762751891828841665, −1.41836281146224207341166714942, 0,
1.41836281146224207341166714942, 2.17354860040762751891828841665, 3.35115029942198380223595347013, 4.04589116505934935133784017202, 4.37446907339856385467029983678, 5.32693634898356568211718896894, 6.07799242721001796498360652148, 6.72454424125334333466695346352, 7.62471887998258366822358681114