L(s) = 1 | + 2-s + 3-s + 4-s − 3.23·5-s + 6-s + 4.47·7-s + 8-s + 9-s − 3.23·10-s − 0.763·11-s + 12-s − 4.47·13-s + 4.47·14-s − 3.23·15-s + 16-s − 4·17-s + 18-s − 7.70·19-s − 3.23·20-s + 4.47·21-s − 0.763·22-s + 23-s + 24-s + 5.47·25-s − 4.47·26-s + 27-s + 4.47·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.44·5-s + 0.408·6-s + 1.69·7-s + 0.353·8-s + 0.333·9-s − 1.02·10-s − 0.230·11-s + 0.288·12-s − 1.24·13-s + 1.19·14-s − 0.835·15-s + 0.250·16-s − 0.970·17-s + 0.235·18-s − 1.76·19-s − 0.723·20-s + 0.975·21-s − 0.162·22-s + 0.208·23-s + 0.204·24-s + 1.09·25-s − 0.877·26-s + 0.192·27-s + 0.845·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.663466658\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.663466658\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 7.70T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 9.23T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 0.763T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−13.20437496468343665858667098765, −12.08893794494749345710454310884, −11.43196640376157802207407586136, −10.46050320145949424277731285914, −8.507778463899568844141389404940, −7.965001313707002290840825421273, −6.89818239213305342054302906759, −4.77283228889444835973926204649, −4.26147070459248842008762577400, −2.40973098544209326818139512558,
2.40973098544209326818139512558, 4.26147070459248842008762577400, 4.77283228889444835973926204649, 6.89818239213305342054302906759, 7.965001313707002290840825421273, 8.507778463899568844141389404940, 10.46050320145949424277731285914, 11.43196640376157802207407586136, 12.08893794494749345710454310884, 13.20437496468343665858667098765