L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s + 7-s − 8-s + 9-s + 2·12-s − 4·13-s − 14-s + 16-s + 6·17-s − 18-s − 2·19-s + 2·21-s − 2·24-s + 4·26-s − 4·27-s + 28-s + 6·29-s − 4·31-s − 32-s − 6·34-s + 36-s − 2·37-s + 2·38-s − 8·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.577·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s + 0.436·21-s − 0.408·24-s + 0.784·26-s − 0.769·27-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s − 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.92895295022713, −14.35415004838895, −14.18120843858814, −13.69057364271465, −12.66759575988296, −12.50760655863588, −11.84585823184490, −11.33422240767362, −10.48565238189661, −10.24437427913259, −9.616989158560158, −9.037420426536510, −8.744288985594174, −7.998637566918170, −7.658481570445999, −7.299000227855195, −6.498221514546552, −5.752340917974148, −5.189491629909670, −4.414148679686017, −3.690348301258547, −2.984895538077002, −2.546720698573449, −1.844728002628330, −1.099188308814597, 0,
1.099188308814597, 1.844728002628330, 2.546720698573449, 2.984895538077002, 3.690348301258547, 4.414148679686017, 5.189491629909670, 5.752340917974148, 6.498221514546552, 7.299000227855195, 7.658481570445999, 7.998637566918170, 8.744288985594174, 9.037420426536510, 9.616989158560158, 10.24437427913259, 10.48565238189661, 11.33422240767362, 11.84585823184490, 12.50760655863588, 12.66759575988296, 13.69057364271465, 14.18120843858814, 14.35415004838895, 14.92895295022713