L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 4.24·11-s − 12-s − 4.82·13-s − 2·14-s + 16-s − 1.17·17-s − 18-s − 2.24·19-s − 2·21-s − 4.24·22-s + 23-s + 24-s + 4.82·26-s − 27-s + 2·28-s − 7.65·29-s + 6·31-s − 32-s − 4.24·33-s + 1.17·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 0.333·9-s + 1.27·11-s − 0.288·12-s − 1.33·13-s − 0.534·14-s + 0.250·16-s − 0.284·17-s − 0.235·18-s − 0.514·19-s − 0.436·21-s − 0.904·22-s + 0.208·23-s + 0.204·24-s + 0.946·26-s − 0.192·27-s + 0.377·28-s − 1.42·29-s + 1.07·31-s − 0.176·32-s − 0.738·33-s + 0.200·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 3.41T + 61T^{2} \) |
| 67 | \( 1 - 0.585T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 3.65T + 73T^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.139417252978483039158086461813, −7.56845724493431727665088476758, −6.71530493635204233734222748307, −6.24071564106165129026177821930, −5.09526678448101388994915688245, −4.55855340168259282939843617172, −3.46715147162141545881031913478, −2.16335458216307610355650323753, −1.38595951800893395693136799332, 0,
1.38595951800893395693136799332, 2.16335458216307610355650323753, 3.46715147162141545881031913478, 4.55855340168259282939843617172, 5.09526678448101388994915688245, 6.24071564106165129026177821930, 6.71530493635204233734222748307, 7.56845724493431727665088476758, 8.139417252978483039158086461813