L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 3·7-s + 8-s + 9-s + 11-s + 12-s − 13-s + 3·14-s + 16-s + 17-s + 18-s + 4·19-s + 3·21-s + 22-s − 5·23-s + 24-s − 26-s + 27-s + 3·28-s − 6·29-s − 8·31-s + 32-s + 33-s + 34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.801·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.654·21-s + 0.213·22-s − 1.04·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.566·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−12.57729630043438, −12.36653414492961, −11.97806335871382, −11.29226493950106, −11.12413552405680, −10.63191043470777, −10.00213700123069, −9.504160342138139, −9.178120663324832, −8.573423666776168, −8.032582875756643, −7.600378846843986, −7.382611286190703, −6.845459810465029, −6.002791768993722, −5.771567230082667, −5.177240061947571, −4.732472338304832, −4.190686638031562, −3.745401875142045, −3.274343656634619, −2.631500469935817, −1.983229123797531, −1.663107052098964, −1.041701201897140, 0,
1.041701201897140, 1.663107052098964, 1.983229123797531, 2.631500469935817, 3.274343656634619, 3.745401875142045, 4.190686638031562, 4.732472338304832, 5.177240061947571, 5.771567230082667, 6.002791768993722, 6.845459810465029, 7.382611286190703, 7.600378846843986, 8.032582875756643, 8.573423666776168, 9.178120663324832, 9.504160342138139, 10.00213700123069, 10.63191043470777, 11.12413552405680, 11.29226493950106, 11.97806335871382, 12.36653414492961, 12.57729630043438