| L(s) = 1 | − 3·9-s − 2·13-s + 17-s − 4·19-s − 8·23-s + 2·29-s − 8·31-s − 2·37-s − 2·41-s − 4·43-s − 6·53-s − 4·59-s + 6·61-s + 4·67-s + 8·71-s + 2·73-s + 9·81-s − 4·83-s + 6·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·117-s + ⋯ |
| L(s) = 1 | − 9-s − 0.554·13-s + 0.242·17-s − 0.917·19-s − 1.66·23-s + 0.371·29-s − 1.43·31-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.824·53-s − 0.520·59-s + 0.768·61-s + 0.488·67-s + 0.949·71-s + 0.234·73-s + 81-s − 0.439·83-s + 0.635·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.554·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.76572539071435, −12.33481083216160, −11.91862079622642, −11.47367139662037, −11.08229949313958, −10.41621313187783, −10.25815956193438, −9.569210439799161, −9.189297551786061, −8.665834674814229, −8.182244490952900, −7.875284057607888, −7.337066425782139, −6.655358554033428, −6.319818314248958, −5.811087490902731, −5.274531472239684, −4.912336722650262, −4.200374105340858, −3.696485155493338, −3.259697153282158, −2.526765284578372, −2.084042783061611, −1.596195452968753, −0.5386178446340935, 0,
0.5386178446340935, 1.596195452968753, 2.084042783061611, 2.526765284578372, 3.259697153282158, 3.696485155493338, 4.200374105340858, 4.912336722650262, 5.274531472239684, 5.811087490902731, 6.319818314248958, 6.655358554033428, 7.337066425782139, 7.875284057607888, 8.182244490952900, 8.665834674814229, 9.189297551786061, 9.569210439799161, 10.25815956193438, 10.41621313187783, 11.08229949313958, 11.47367139662037, 11.91862079622642, 12.33481083216160, 12.76572539071435
