L(s) = 1 | − 2-s − 3-s + 4-s − 1.35·5-s + 6-s + 7-s − 8-s + 9-s + 1.35·10-s + 4.29·11-s − 12-s − 14-s + 1.35·15-s + 16-s + 2.66·17-s − 18-s + 4.29·19-s − 1.35·20-s − 21-s − 4.29·22-s − 4.63·23-s + 24-s − 3.15·25-s − 27-s + 28-s + 5.54·29-s − 1.35·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.606·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.429·10-s + 1.29·11-s − 0.288·12-s − 0.267·14-s + 0.350·15-s + 0.250·16-s + 0.646·17-s − 0.235·18-s + 0.985·19-s − 0.303·20-s − 0.218·21-s − 0.915·22-s − 0.965·23-s + 0.204·24-s − 0.631·25-s − 0.192·27-s + 0.188·28-s + 1.02·29-s − 0.247·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 1.35T + 5T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 17 | \( 1 - 2.66T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 + 5.51T + 31T^{2} \) |
| 37 | \( 1 + 3.53T + 37T^{2} \) |
| 41 | \( 1 + 9.45T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 - 4.87T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 - 5.62T + 83T^{2} \) |
| 89 | \( 1 - 9.81T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−7.59231147320446047801469069707, −6.97366863515197448246347254149, −6.35551405041543986268244543576, −5.54642499062356863897999221591, −4.81391844280506651011488316962, −3.82777885143691531009803757130, −3.31734097557267469371728621998, −1.86871805689286513560200515065, −1.19560434175911772478538194288, 0,
1.19560434175911772478538194288, 1.86871805689286513560200515065, 3.31734097557267469371728621998, 3.82777885143691531009803757130, 4.81391844280506651011488316962, 5.54642499062356863897999221591, 6.35551405041543986268244543576, 6.97366863515197448246347254149, 7.59231147320446047801469069707