L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 2·7-s − 2·9-s + 2·10-s − 2·12-s + 4·14-s + 15-s − 4·16-s + 2·17-s + 4·18-s − 2·20-s + 2·21-s − 23-s − 4·25-s + 5·27-s − 4·28-s − 2·30-s − 7·31-s + 8·32-s − 4·34-s + 2·35-s − 4·36-s − 3·37-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 2/3·9-s + 0.632·10-s − 0.577·12-s + 1.06·14-s + 0.258·15-s − 16-s + 0.485·17-s + 0.942·18-s − 0.447·20-s + 0.436·21-s − 0.208·23-s − 4/5·25-s + 0.962·27-s − 0.755·28-s − 0.365·30-s − 1.25·31-s + 1.41·32-s − 0.685·34-s + 0.338·35-s − 2/3·36-s − 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−16.33876970759384, −15.94136401912385, −15.36079251592833, −14.69187686722283, −13.92654662376284, −13.55158006948452, −12.55692172152170, −12.33332213238782, −11.48257315472483, −11.17381612847762, −10.61077523102692, −9.997803318826309, −9.470201127780350, −9.047361338686055, −8.217506491730922, −7.972000332755686, −7.196250039602744, −6.644997946132623, −6.035780196715335, −5.348640601367650, −4.615561361841008, −3.663131710551717, −3.082857453782681, −2.079764001679285, −1.244472589701135, 0, 0,
1.244472589701135, 2.079764001679285, 3.082857453782681, 3.663131710551717, 4.615561361841008, 5.348640601367650, 6.035780196715335, 6.644997946132623, 7.196250039602744, 7.972000332755686, 8.217506491730922, 9.047361338686055, 9.470201127780350, 9.997803318826309, 10.61077523102692, 11.17381612847762, 11.48257315472483, 12.33332213238782, 12.55692172152170, 13.55158006948452, 13.92654662376284, 14.69187686722283, 15.36079251592833, 15.94136401912385, 16.33876970759384