L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s + 13-s + 16-s + 6·17-s − 20-s − 22-s + 4·23-s + 25-s − 26-s − 29-s − 2·31-s − 32-s − 6·34-s + 4·37-s + 40-s − 5·41-s + 6·43-s + 44-s − 4·46-s − 7·49-s − 50-s + 52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.277·13-s + 1/4·16-s + 1.45·17-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.185·29-s − 0.359·31-s − 0.176·32-s − 1.02·34-s + 0.657·37-s + 0.158·40-s − 0.780·41-s + 0.914·43-s + 0.150·44-s − 0.589·46-s − 49-s − 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.65114261525732, −12.25224208761645, −11.69034000224987, −11.44790503813717, −10.93475031184102, −10.37980218666443, −10.15473779622751, −9.449413108382172, −9.122885102491416, −8.774028041938956, −8.019964046056313, −7.826228821676167, −7.373003806457338, −6.877939048485789, −6.245688323104278, −5.960552505248451, −5.242871632421849, −4.852320972461252, −4.146726216366095, −3.586526164197242, −3.136652342782636, −2.687984020426891, −1.811976633479667, −1.313312492326368, −0.7855459582676188, 0,
0.7855459582676188, 1.313312492326368, 1.811976633479667, 2.687984020426891, 3.136652342782636, 3.586526164197242, 4.146726216366095, 4.852320972461252, 5.242871632421849, 5.960552505248451, 6.245688323104278, 6.877939048485789, 7.373003806457338, 7.826228821676167, 8.019964046056313, 8.774028041938956, 9.122885102491416, 9.449413108382172, 10.15473779622751, 10.37980218666443, 10.93475031184102, 11.44790503813717, 11.69034000224987, 12.25224208761645, 12.65114261525732