L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 7-s + 3·8-s + 9-s − 12-s − 2·13-s − 14-s − 16-s + 3·17-s − 18-s + 3·19-s + 21-s + 23-s + 3·24-s + 2·26-s + 27-s − 28-s − 6·29-s + 2·31-s − 5·32-s − 3·34-s − 36-s − 3·37-s − 3·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.688·19-s + 0.218·21-s + 0.208·23-s + 0.612·24-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.359·31-s − 0.883·32-s − 0.514·34-s − 1/6·36-s − 0.493·37-s − 0.486·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−7.43317210880166113711428075594, −7.18490297160319187246232479017, −6.03070698555280396209633224480, −5.08377463017473135012958569593, −4.74624890205055653165753794090, −3.70927620102147368308246941077, −3.11601675359563286821948628943, −1.92887542151944080849157361852, −1.25926769222626214579270656188, 0,
1.25926769222626214579270656188, 1.92887542151944080849157361852, 3.11601675359563286821948628943, 3.70927620102147368308246941077, 4.74624890205055653165753794090, 5.08377463017473135012958569593, 6.03070698555280396209633224480, 7.18490297160319187246232479017, 7.43317210880166113711428075594