| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 2·7-s + 8-s + 9-s − 2·10-s − 6·11-s − 12-s + 13-s + 2·14-s + 2·15-s + 16-s + 3·17-s + 18-s − 6·19-s − 2·20-s − 2·21-s − 6·22-s + 23-s − 24-s − 25-s + 26-s − 27-s + 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.80·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 1.37·19-s − 0.447·20-s − 0.436·21-s − 1.27·22-s + 0.208·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.94366021846652, −13.23899662036929, −13.06303285758476, −12.62103066396797, −12.04463765737934, −11.46702951878261, −11.19147931521195, −10.77856877107009, −10.27764463352681, −9.794127250445113, −8.895220436291807, −8.284013969883161, −7.873691504733430, −7.516270077829493, −7.037423779838183, −6.193850509441654, −5.757172758356286, −5.238902636587102, −4.765357821513146, −4.294795261418342, −3.615959139846408, −3.165628662999580, −2.179561226907387, −1.884665755069365, −0.7410755054625090, 0,
0.7410755054625090, 1.884665755069365, 2.179561226907387, 3.165628662999580, 3.615959139846408, 4.294795261418342, 4.765357821513146, 5.238902636587102, 5.757172758356286, 6.193850509441654, 7.037423779838183, 7.516270077829493, 7.873691504733430, 8.284013969883161, 8.895220436291807, 9.794127250445113, 10.27764463352681, 10.77856877107009, 11.19147931521195, 11.46702951878261, 12.04463765737934, 12.62103066396797, 13.06303285758476, 13.23899662036929, 13.94366021846652
