L(s) = 1 | − 2.28·2-s − 3-s + 3.21·4-s + 1.64·5-s + 2.28·6-s + 3.93·7-s − 2.77·8-s + 9-s − 3.75·10-s − 0.343·11-s − 3.21·12-s + 0.744·13-s − 8.98·14-s − 1.64·15-s − 0.0902·16-s − 17-s − 2.28·18-s − 7.57·19-s + 5.28·20-s − 3.93·21-s + 0.784·22-s − 3.77·23-s + 2.77·24-s − 2.29·25-s − 1.70·26-s − 27-s + 12.6·28-s + ⋯ |
L(s) = 1 | − 1.61·2-s − 0.577·3-s + 1.60·4-s + 0.734·5-s + 0.932·6-s + 1.48·7-s − 0.981·8-s + 0.333·9-s − 1.18·10-s − 0.103·11-s − 0.928·12-s + 0.206·13-s − 2.40·14-s − 0.424·15-s − 0.0225·16-s − 0.242·17-s − 0.538·18-s − 1.73·19-s + 1.18·20-s − 0.858·21-s + 0.167·22-s − 0.787·23-s + 0.566·24-s − 0.459·25-s − 0.333·26-s − 0.192·27-s + 2.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 + 0.343T + 11T^{2} \) |
| 13 | \( 1 - 0.744T + 13T^{2} \) |
| 19 | \( 1 + 7.57T + 19T^{2} \) |
| 23 | \( 1 + 3.77T + 23T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 - 4.72T + 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 43 | \( 1 + 0.185T + 43T^{2} \) |
| 47 | \( 1 - 8.35T + 47T^{2} \) |
| 53 | \( 1 - 8.10T + 53T^{2} \) |
| 59 | \( 1 + 7.50T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 + 1.23T + 71T^{2} \) |
| 73 | \( 1 + 9.29T + 73T^{2} \) |
| 79 | \( 1 - 9.60T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 3.05T + 89T^{2} \) |
| 97 | \( 1 + 6.82T + 97T^{2} \) |
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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
−7.83511557860896591292948368370, −6.92192368168901449044645374031, −6.25574588873903996734353784732, −5.65229416526977884067217638415, −4.69993993396808027217479635723, −4.11311319224142896742995001935, −2.42354946533342827576214980346, −1.90736662720583020937389419265, −1.21016372259396342102788555266, 0,
1.21016372259396342102788555266, 1.90736662720583020937389419265, 2.42354946533342827576214980346, 4.11311319224142896742995001935, 4.69993993396808027217479635723, 5.65229416526977884067217638415, 6.25574588873903996734353784732, 6.92192368168901449044645374031, 7.83511557860896591292948368370