L(s) = 1 | − 2·3-s − 2·4-s − 3·5-s + 9-s + 3·11-s + 4·12-s − 4·13-s + 6·15-s + 4·16-s + 3·17-s + 6·20-s + 4·25-s + 4·27-s − 6·29-s − 4·31-s − 6·33-s − 2·36-s − 2·37-s + 8·39-s − 6·41-s − 43-s − 6·44-s − 3·45-s + 3·47-s − 8·48-s − 6·51-s + 8·52-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 4-s − 1.34·5-s + 1/3·9-s + 0.904·11-s + 1.15·12-s − 1.10·13-s + 1.54·15-s + 16-s + 0.727·17-s + 1.34·20-s + 4/5·25-s + 0.769·27-s − 1.11·29-s − 0.718·31-s − 1.04·33-s − 1/3·36-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 0.152·43-s − 0.904·44-s − 0.447·45-s + 0.437·47-s − 1.15·48-s − 0.840·51-s + 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 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 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.24062917513488, −15.58788124547474, −14.97625082444832, −14.40949841703842, −14.15230409835509, −13.14532597696508, −12.51359672009654, −12.25815090454689, −11.64927693836599, −11.32739203066337, −10.58331962908708, −9.957876800900652, −9.358863964568489, −8.757498864886458, −8.082079215054112, −7.468552785540221, −7.015775804758812, −6.103219812061688, −5.528913749029299, −4.833805940802903, −4.465478507359845, −3.637043445132515, −3.209213954492848, −1.704338888578465, −0.6443735371877729, 0,
0.6443735371877729, 1.704338888578465, 3.209213954492848, 3.637043445132515, 4.465478507359845, 4.833805940802903, 5.528913749029299, 6.103219812061688, 7.015775804758812, 7.468552785540221, 8.082079215054112, 8.757498864886458, 9.358863964568489, 9.957876800900652, 10.58331962908708, 11.32739203066337, 11.64927693836599, 12.25815090454689, 12.51359672009654, 13.14532597696508, 14.15230409835509, 14.40949841703842, 14.97625082444832, 15.58788124547474, 16.24062917513488