| L(s) = 1 | − 2·3-s − 2·4-s + 5-s + 5·7-s + 9-s − 11-s + 4·12-s − 4·13-s − 2·15-s + 4·16-s − 17-s − 4·19-s − 2·20-s − 10·21-s + 6·23-s + 25-s + 4·27-s − 10·28-s − 3·29-s + 2·31-s + 2·33-s + 5·35-s − 2·36-s + 8·37-s + 8·39-s + 9·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 0.447·5-s + 1.88·7-s + 1/3·9-s − 0.301·11-s + 1.15·12-s − 1.10·13-s − 0.516·15-s + 16-s − 0.242·17-s − 0.917·19-s − 0.447·20-s − 2.18·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.88·28-s − 0.557·29-s + 0.359·31-s + 0.348·33-s + 0.845·35-s − 1/3·36-s + 1.31·37-s + 1.28·39-s + 1.40·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9641366883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9641366883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−10.17849480089901144304272348460, −9.257661040186113627072300512339, −8.358976954769995768226444756755, −7.66639240851565724305175638479, −6.46722355144798522860208878411, −5.23945127542030902400382749798, −5.09286497121302525229336107198, −4.26090192856892493958348289146, −2.32301448348933296237105671366, −0.847723027362305869576784954111,
0.847723027362305869576784954111, 2.32301448348933296237105671366, 4.26090192856892493958348289146, 5.09286497121302525229336107198, 5.23945127542030902400382749798, 6.46722355144798522860208878411, 7.66639240851565724305175638479, 8.358976954769995768226444756755, 9.257661040186113627072300512339, 10.17849480089901144304272348460
