L(s) = 1 | − 2-s + 3-s − 4-s − 3·5-s − 6-s + 7-s + 3·8-s + 9-s + 3·10-s + 6·11-s − 12-s − 6·13-s − 14-s − 3·15-s − 16-s − 17-s − 18-s + 6·19-s + 3·20-s + 21-s − 6·22-s + 6·23-s + 3·24-s + 4·25-s + 6·26-s + 27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.948·10-s + 1.80·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.774·15-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.670·20-s + 0.218·21-s − 1.27·22-s + 1.25·23-s + 0.612·24-s + 4/5·25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.38808451100912, −15.33612446962717, −14.94408679964406, −14.70380073178517, −13.84826962625982, −13.62378814994830, −12.64592287768261, −12.12618412733689, −11.52759362501634, −11.36128477132887, −10.23096471564520, −9.746544633216211, −9.354067503967411, −8.580406803591663, −8.349017125213281, −7.589175040037986, −7.081207860207030, −6.792223529995703, −5.352220214682454, −4.682026531343758, −4.344681123606651, −3.518671407075754, −2.983820556084452, −1.694344503381537, −1.035581937693167, 0,
1.035581937693167, 1.694344503381537, 2.983820556084452, 3.518671407075754, 4.344681123606651, 4.682026531343758, 5.352220214682454, 6.792223529995703, 7.081207860207030, 7.589175040037986, 8.349017125213281, 8.580406803591663, 9.354067503967411, 9.746544633216211, 10.23096471564520, 11.36128477132887, 11.52759362501634, 12.12618412733689, 12.64592287768261, 13.62378814994830, 13.84826962625982, 14.70380073178517, 14.94408679964406, 15.33612446962717, 16.38808451100912