| L(s) = 1 | − 3-s − 2·4-s + 4·7-s − 2·9-s − 6·11-s + 2·12-s − 13-s + 4·16-s − 6·17-s − 4·19-s − 4·21-s − 3·23-s + 5·27-s − 8·28-s − 3·29-s − 4·31-s + 6·33-s + 4·36-s − 2·37-s + 39-s + 6·41-s + 7·43-s + 12·44-s − 4·48-s + 9·49-s + 6·51-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1.51·7-s − 2/3·9-s − 1.80·11-s + 0.577·12-s − 0.277·13-s + 16-s − 1.45·17-s − 0.917·19-s − 0.872·21-s − 0.625·23-s + 0.962·27-s − 1.51·28-s − 0.557·29-s − 0.718·31-s + 1.04·33-s + 2/3·36-s − 0.328·37-s + 0.160·39-s + 0.937·41-s + 1.06·43-s + 1.80·44-s − 0.577·48-s + 9/7·49-s + 0.840·51-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−10.85194742515835258059526457232, −10.65703411077808150300539182298, −9.088843841365897209974292169642, −8.314976154876773643802249838060, −7.58470653085664427086815596244, −5.84441229372916213818129575211, −5.07836611320697081751291448911, −4.33348131529579942321198409567, −2.32784842593438550687981568602, 0,
2.32784842593438550687981568602, 4.33348131529579942321198409567, 5.07836611320697081751291448911, 5.84441229372916213818129575211, 7.58470653085664427086815596244, 8.314976154876773643802249838060, 9.088843841365897209974292169642, 10.65703411077808150300539182298, 10.85194742515835258059526457232
