| L(s) = 1 | − 5-s + 3·11-s + 6·13-s − 5·17-s + 19-s − 7·23-s − 4·25-s − 2·29-s − 5·31-s + 3·37-s − 2·41-s + 4·43-s − 5·47-s + 53-s − 3·55-s − 15·59-s + 5·61-s − 6·65-s + 9·67-s − 7·73-s − 79-s − 12·83-s + 5·85-s + 7·89-s − 95-s + 2·97-s + 3·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.904·11-s + 1.66·13-s − 1.21·17-s + 0.229·19-s − 1.45·23-s − 4/5·25-s − 0.371·29-s − 0.898·31-s + 0.493·37-s − 0.312·41-s + 0.609·43-s − 0.729·47-s + 0.137·53-s − 0.404·55-s − 1.95·59-s + 0.640·61-s − 0.744·65-s + 1.09·67-s − 0.819·73-s − 0.112·79-s − 1.31·83-s + 0.542·85-s + 0.741·89-s − 0.102·95-s + 0.203·97-s + 0.298·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−7.68028487353977020685434420935, −6.81299387469983938345337002747, −6.18283629506126241248271512806, −5.69553024467688056130746201791, −4.48722885144786324382387208225, −3.93920665375416635707988839718, −3.41923492434141787529341330204, −2.13434730284385344198940692364, −1.34283087210270614855046078004, 0,
1.34283087210270614855046078004, 2.13434730284385344198940692364, 3.41923492434141787529341330204, 3.93920665375416635707988839718, 4.48722885144786324382387208225, 5.69553024467688056130746201791, 6.18283629506126241248271512806, 6.81299387469983938345337002747, 7.68028487353977020685434420935
