L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 2·11-s − 5·13-s − 14-s + 16-s − 5·17-s + 8·19-s − 20-s − 2·22-s − 23-s + 25-s + 5·26-s + 28-s − 9·29-s + 31-s − 32-s + 5·34-s − 35-s − 8·37-s − 8·38-s + 40-s − 2·41-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 + 0.603·11-s − 1.38·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s + 1.83·19-s − 0.223·20-s − 0.426·22-s − 0.208·23-s + 1/5·25-s + 0.980·26-s + 0.188·28-s − 1.67·29-s + 0.179·31-s − 0.176·32-s + 0.857·34-s − 0.169·35-s − 1.31·37-s − 1.29·38-s + 0.158·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.114599065981276406214862423910, −7.907272629442069021516624708400, −7.41342827773699841846370102608, −6.77350070376964788698840394975, −5.60881433440025361728359489120, −4.77023759749701418451481440875, −3.74046525828913219251762263321, −2.62644902559356869926653174395, −1.52195069046361628471817715823, 0,
1.52195069046361628471817715823, 2.62644902559356869926653174395, 3.74046525828913219251762263321, 4.77023759749701418451481440875, 5.60881433440025361728359489120, 6.77350070376964788698840394975, 7.41342827773699841846370102608, 7.907272629442069021516624708400, 9.114599065981276406214862423910