L(s) = 1 | + 3-s − 7-s + 9-s − 11-s + 2·13-s − 17-s + 7·19-s − 21-s + 5·23-s + 27-s + 3·29-s + 4·31-s − 33-s + 2·37-s + 2·39-s − 12·41-s − 43-s − 4·47-s + 49-s − 51-s + 53-s + 7·57-s − 9·59-s − 11·61-s − 63-s + 2·67-s + 5·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.242·17-s + 1.60·19-s − 0.218·21-s + 1.04·23-s + 0.192·27-s + 0.557·29-s + 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.320·39-s − 1.87·41-s − 0.152·43-s − 0.583·47-s + 1/7·49-s − 0.140·51-s + 0.137·53-s + 0.927·57-s − 1.17·59-s − 1.40·61-s − 0.125·63-s + 0.244·67-s + 0.601·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.93313514950750, −13.51345649375307, −13.36043211733128, −12.67859904546311, −12.08563572345512, −11.73633903633723, −11.06916016021960, −10.61884935578259, −10.01433788330860, −9.591913660219178, −9.150422264995458, −8.523489775156005, −8.163566867467800, −7.538215351507101, −7.014031728890989, −6.559104902067825, −5.940327519294265, −5.245802148484235, −4.805422577684341, −4.163337868465072, −3.234537759511368, −3.199950593914747, −2.486467834846924, −1.518681850686104, −1.070491344638488, 0,
1.070491344638488, 1.518681850686104, 2.486467834846924, 3.199950593914747, 3.234537759511368, 4.163337868465072, 4.805422577684341, 5.245802148484235, 5.940327519294265, 6.559104902067825, 7.014031728890989, 7.538215351507101, 8.163566867467800, 8.523489775156005, 9.150422264995458, 9.591913660219178, 10.01433788330860, 10.61884935578259, 11.06916016021960, 11.73633903633723, 12.08563572345512, 12.67859904546311, 13.36043211733128, 13.51345649375307, 13.93313514950750