L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 2·11-s − 13-s − 14-s + 16-s − 3·17-s + 2·22-s + 23-s + 26-s + 28-s + 5·29-s + 7·31-s − 32-s + 3·34-s − 2·37-s − 7·41-s − 11·43-s − 2·44-s − 46-s − 8·47-s + 49-s − 52-s + 53-s − 56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.426·22-s + 0.208·23-s + 0.196·26-s + 0.188·28-s + 0.928·29-s + 1.25·31-s − 0.176·32-s + 0.514·34-s − 0.328·37-s − 1.09·41-s − 1.67·43-s − 0.301·44-s − 0.147·46-s − 1.16·47-s + 1/7·49-s − 0.138·52-s + 0.137·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.366632324211996219991005794548, −7.73331032905174458096636375950, −6.85137733747305961108725942830, −6.29178096130104433835567544947, −5.15483028346137606042980515566, −4.58607310529684007211324334325, −3.29455399221364617927893806715, −2.44116661444188442116718256018, −1.41570341834852844853092885995, 0,
1.41570341834852844853092885995, 2.44116661444188442116718256018, 3.29455399221364617927893806715, 4.58607310529684007211324334325, 5.15483028346137606042980515566, 6.29178096130104433835567544947, 6.85137733747305961108725942830, 7.73331032905174458096636375950, 8.366632324211996219991005794548