| L(s) = 1 | + 20·3-s − 24·7-s + 157·9-s − 124·11-s − 478·13-s + 1.19e3·17-s − 3.04e3·19-s − 480·21-s + 184·23-s − 1.72e3·27-s − 3.28e3·29-s + 5.72e3·31-s − 2.48e3·33-s − 1.03e4·37-s − 9.56e3·39-s − 8.88e3·41-s − 9.18e3·43-s + 2.36e4·47-s − 1.62e4·49-s + 2.39e4·51-s − 1.16e4·53-s − 6.08e4·57-s − 1.68e4·59-s − 1.84e4·61-s − 3.76e3·63-s − 1.55e4·67-s + 3.68e3·69-s + ⋯ |
| L(s) = 1 | + 1.28·3-s − 0.185·7-s + 0.646·9-s − 0.308·11-s − 0.784·13-s + 1.00·17-s − 1.93·19-s − 0.237·21-s + 0.0725·23-s − 0.454·27-s − 0.724·29-s + 1.07·31-s − 0.396·33-s − 1.24·37-s − 1.00·39-s − 0.825·41-s − 0.757·43-s + 1.56·47-s − 0.965·49-s + 1.28·51-s − 0.571·53-s − 2.48·57-s − 0.631·59-s − 0.635·61-s − 0.119·63-s − 0.422·67-s + 0.0930·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 20 T + p^{5} T^{2} \) |
| 7 | \( 1 + 24 T + p^{5} T^{2} \) |
| 11 | \( 1 + 124 T + p^{5} T^{2} \) |
| 13 | \( 1 + 478 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1198 T + p^{5} T^{2} \) |
| 19 | \( 1 + 3044 T + p^{5} T^{2} \) |
| 23 | \( 1 - 8 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 3282 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5728 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10326 T + p^{5} T^{2} \) |
| 41 | \( 1 + 8886 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9188 T + p^{5} T^{2} \) |
| 47 | \( 1 - 23664 T + p^{5} T^{2} \) |
| 53 | \( 1 + 11686 T + p^{5} T^{2} \) |
| 59 | \( 1 + 16876 T + p^{5} T^{2} \) |
| 61 | \( 1 + 18482 T + p^{5} T^{2} \) |
| 67 | \( 1 + 15532 T + p^{5} T^{2} \) |
| 71 | \( 1 - 31960 T + p^{5} T^{2} \) |
| 73 | \( 1 - 4886 T + p^{5} T^{2} \) |
| 79 | \( 1 + 44560 T + p^{5} T^{2} \) |
| 83 | \( 1 - 67364 T + p^{5} T^{2} \) |
| 89 | \( 1 - 71994 T + p^{5} T^{2} \) |
| 97 | \( 1 + 48866 T + p^{5} T^{2} \) |
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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
−9.905861510749365055372223221826, −8.973303703289617269063845281216, −8.226232904688023965030511384361, −7.45277062603882917066546078671, −6.31224863189422721822346963444, −4.97800704004705923060636845250, −3.76869845915186119685665845211, −2.79404250010288554206252306807, −1.83245996031274823478267151935, 0,
1.83245996031274823478267151935, 2.79404250010288554206252306807, 3.76869845915186119685665845211, 4.97800704004705923060636845250, 6.31224863189422721822346963444, 7.45277062603882917066546078671, 8.226232904688023965030511384361, 8.973303703289617269063845281216, 9.905861510749365055372223221826
