| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s + 15-s + 2·17-s + 2·19-s + 21-s − 7·23-s − 4·25-s + 5·27-s + 10·29-s + 7·31-s + 35-s − 9·37-s + 2·41-s + 4·43-s + 2·45-s + 8·47-s + 49-s − 2·51-s + 2·53-s − 2·57-s − 15·59-s + 14·61-s + 2·63-s + 3·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.258·15-s + 0.485·17-s + 0.458·19-s + 0.218·21-s − 1.45·23-s − 4/5·25-s + 0.962·27-s + 1.85·29-s + 1.25·31-s + 0.169·35-s − 1.47·37-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 0.264·57-s − 1.95·59-s + 1.79·61-s + 0.251·63-s + 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 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
−7.64944230568472102259411594799, −6.85353459584934900980455616577, −6.09512858858467819844874020708, −5.67357727613238102864894674620, −4.77658343752724850006675096838, −4.02938013783735219004396113744, −3.18030571310674762768315503659, −2.40641604963538889234402285810, −1.04449595947954283209629980492, 0,
1.04449595947954283209629980492, 2.40641604963538889234402285810, 3.18030571310674762768315503659, 4.02938013783735219004396113744, 4.77658343752724850006675096838, 5.67357727613238102864894674620, 6.09512858858467819844874020708, 6.85353459584934900980455616577, 7.64944230568472102259411594799
