L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 5·11-s − 12-s − 3·13-s + 16-s − 6·17-s − 18-s − 19-s − 5·22-s + 23-s + 24-s + 3·26-s − 27-s − 6·29-s − 7·31-s − 32-s − 5·33-s + 6·34-s + 36-s + 6·37-s + 38-s + 3·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.50·11-s − 0.288·12-s − 0.832·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.229·19-s − 1.06·22-s + 0.208·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s − 1.11·29-s − 1.25·31-s − 0.176·32-s − 0.870·33-s + 1.02·34-s + 1/6·36-s + 0.986·37-s + 0.162·38-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.63515220787859, −12.92638163389414, −12.72154924533250, −12.07584304235096, −11.52164942764271, −11.33008591028032, −10.87032680309218, −10.24325271845979, −9.762080115873641, −9.166048160215933, −9.041860653344660, −8.422203969878518, −7.705062040709167, −7.150234630998207, −6.859596406221613, −6.358910310426707, −5.832513215089538, −5.198935685300526, −4.619186656040628, −3.961803802766199, −3.621431961140474, −2.614531419833358, −2.038007137082537, −1.543234701971718, −0.6952204384174193, 0,
0.6952204384174193, 1.543234701971718, 2.038007137082537, 2.614531419833358, 3.621431961140474, 3.961803802766199, 4.619186656040628, 5.198935685300526, 5.832513215089538, 6.358910310426707, 6.859596406221613, 7.150234630998207, 7.705062040709167, 8.422203969878518, 9.041860653344660, 9.166048160215933, 9.762080115873641, 10.24325271845979, 10.87032680309218, 11.33008591028032, 11.52164942764271, 12.07584304235096, 12.72154924533250, 12.92638163389414, 13.63515220787859