L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 4·13-s + 14-s + 16-s + 2·19-s + 20-s − 23-s + 25-s − 4·26-s + 28-s − 6·29-s + 2·31-s + 32-s + 35-s − 10·37-s + 2·38-s + 40-s − 6·41-s − 4·43-s − 46-s − 6·47-s + 49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.458·19-s + 0.223·20-s − 0.208·23-s + 1/5·25-s − 0.784·26-s + 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s + 0.169·35-s − 1.64·37-s + 0.324·38-s + 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.147·46-s − 0.875·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−16.41664104528539, −15.62338067303976, −15.17550517663952, −14.69674704552395, −14.07958455105495, −13.66333859798755, −13.14066157947187, −12.35250893326888, −12.03566816624502, −11.46095845591734, −10.67578009171494, −10.26204514902243, −9.537827778192677, −9.032270035168255, −8.113791208163789, −7.615947600682880, −6.905543871538469, −6.404838409821572, −5.454454814643366, −5.161767834204562, −4.493320682283809, −3.611238935726512, −2.950994222973927, −2.075865807623234, −1.483459387059458, 0,
1.483459387059458, 2.075865807623234, 2.950994222973927, 3.611238935726512, 4.493320682283809, 5.161767834204562, 5.454454814643366, 6.404838409821572, 6.905543871538469, 7.615947600682880, 8.113791208163789, 9.032270035168255, 9.537827778192677, 10.26204514902243, 10.67578009171494, 11.46095845591734, 12.03566816624502, 12.35250893326888, 13.14066157947187, 13.66333859798755, 14.07958455105495, 14.69674704552395, 15.17550517663952, 15.62338067303976, 16.41664104528539