| L(s) = 1 | − 5-s − 2·11-s + 13-s + 2·17-s − 2·19-s − 8·23-s + 25-s − 2·31-s − 2·37-s − 2·41-s + 12·43-s − 12·47-s − 6·53-s + 2·55-s − 12·59-s − 8·61-s − 65-s + 12·67-s − 6·71-s − 14·73-s − 16·79-s − 2·85-s − 6·89-s + 2·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s + 0.277·13-s + 0.485·17-s − 0.458·19-s − 1.66·23-s + 1/5·25-s − 0.359·31-s − 0.328·37-s − 0.312·41-s + 1.82·43-s − 1.75·47-s − 0.824·53-s + 0.269·55-s − 1.56·59-s − 1.02·61-s − 0.124·65-s + 1.46·67-s − 0.712·71-s − 1.63·73-s − 1.80·79-s − 0.216·85-s − 0.635·89-s + 0.205·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.27968546127175, −12.94178155278278, −12.36321256834255, −12.13191496158682, −11.54414484636460, −11.00128819111018, −10.71954428820911, −10.10712340184609, −9.755972877528338, −9.199116628476602, −8.594832416796834, −8.187048443172655, −7.736815644025811, −7.412277490128775, −6.705138942683270, −6.186285887684965, −5.748284885733824, −5.266070188650263, −4.490031646451858, −4.243245529243917, −3.569458941040298, −3.032754997085745, −2.494265214592821, −1.714642611728736, −1.260851324199778, 0, 0,
1.260851324199778, 1.714642611728736, 2.494265214592821, 3.032754997085745, 3.569458941040298, 4.243245529243917, 4.490031646451858, 5.266070188650263, 5.748284885733824, 6.186285887684965, 6.705138942683270, 7.412277490128775, 7.736815644025811, 8.187048443172655, 8.594832416796834, 9.199116628476602, 9.755972877528338, 10.10712340184609, 10.71954428820911, 11.00128819111018, 11.54414484636460, 12.13191496158682, 12.36321256834255, 12.94178155278278, 13.27968546127175
