| L(s) = 1 | + 3-s − 2·4-s − 5-s − 7-s − 2·9-s − 2·12-s − 5·13-s − 15-s + 4·16-s − 3·17-s − 2·19-s + 2·20-s − 21-s − 6·23-s + 25-s − 5·27-s + 2·28-s − 3·29-s − 4·31-s + 35-s + 4·36-s + 2·37-s − 5·39-s + 12·41-s + 10·43-s + 2·45-s + 9·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.577·12-s − 1.38·13-s − 0.258·15-s + 16-s − 0.727·17-s − 0.458·19-s + 0.447·20-s − 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.962·27-s + 0.377·28-s − 0.557·29-s − 0.718·31-s + 0.169·35-s + 2/3·36-s + 0.328·37-s − 0.800·39-s + 1.87·41-s + 1.52·43-s + 0.298·45-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6648458645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6648458645\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.504764710422860488143194115790, −7.67950449335459092353880992506, −7.28726628142540726038141332638, −5.98086532326102961421297628547, −5.49671842174989089979243313601, −4.26526382517520030491171627830, −4.09257990991624762122042789987, −2.90308901018164947604047652054, −2.20945739428189555232801439935, −0.42426621729141121803731134362,
0.42426621729141121803731134362, 2.20945739428189555232801439935, 2.90308901018164947604047652054, 4.09257990991624762122042789987, 4.26526382517520030491171627830, 5.49671842174989089979243313601, 5.98086532326102961421297628547, 7.28726628142540726038141332638, 7.67950449335459092353880992506, 8.504764710422860488143194115790
