| L(s) = 1 | + 5-s + 2·7-s − 4·13-s − 6·17-s + 2·19-s + 8·23-s + 25-s + 6·29-s + 2·35-s + 6·37-s − 10·41-s − 2·43-s + 12·47-s − 3·49-s − 6·53-s − 8·59-s − 4·65-s − 4·67-s − 12·71-s − 4·73-s + 10·79-s + 4·83-s − 6·85-s + 14·89-s − 8·91-s + 2·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 1.10·13-s − 1.45·17-s + 0.458·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.338·35-s + 0.986·37-s − 1.56·41-s − 0.304·43-s + 1.75·47-s − 3/7·49-s − 0.824·53-s − 1.04·59-s − 0.496·65-s − 0.488·67-s − 1.42·71-s − 0.468·73-s + 1.12·79-s + 0.439·83-s − 0.650·85-s + 1.48·89-s − 0.838·91-s + 0.205·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.933255188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.933255188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.54632503245624, −12.03892870024619, −11.62522499533056, −11.24975332459216, −10.68099492795587, −10.36375782358687, −9.874924757188545, −9.248524692094034, −8.874480605965453, −8.669092131538460, −7.788016108742215, −7.556428009873899, −7.040417776729923, −6.415310444980477, −6.229736144054121, −5.306083001683921, −4.970691956350152, −4.654587423558703, −4.214524161071322, −3.218568216706878, −2.927275689802014, −2.249036702454221, −1.831055350244944, −1.113958835709395, −0.4659155144989484,
0.4659155144989484, 1.113958835709395, 1.831055350244944, 2.249036702454221, 2.927275689802014, 3.218568216706878, 4.214524161071322, 4.654587423558703, 4.970691956350152, 5.306083001683921, 6.229736144054121, 6.415310444980477, 7.040417776729923, 7.556428009873899, 7.788016108742215, 8.669092131538460, 8.874480605965453, 9.248524692094034, 9.874924757188545, 10.36375782358687, 10.68099492795587, 11.24975332459216, 11.62522499533056, 12.03892870024619, 12.54632503245624
