| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 4·11-s − 12-s + 14-s + 15-s + 16-s + 2·17-s − 18-s + 4·19-s − 20-s + 21-s + 4·22-s − 8·23-s + 24-s + 25-s − 27-s − 28-s + 6·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.48378012266740, −14.90079689118447, −14.01884495690288, −13.63212824432603, −13.06962622319046, −12.25006588780148, −11.92354762278580, −11.71865006810371, −10.67837706432194, −10.49004499652758, −9.865891418256402, −9.602656369073031, −8.549086820046890, −8.196468085064882, −7.691866898838172, −7.132224103846280, −6.487233379702294, −5.901451187487693, −5.351911374973982, −4.662991491974496, −3.968539043005460, −3.071426239067502, −2.650019876128833, −1.633280797960132, −0.7637612112489709, 0,
0.7637612112489709, 1.633280797960132, 2.650019876128833, 3.071426239067502, 3.968539043005460, 4.662991491974496, 5.351911374973982, 5.901451187487693, 6.487233379702294, 7.132224103846280, 7.691866898838172, 8.196468085064882, 8.549086820046890, 9.602656369073031, 9.865891418256402, 10.49004499652758, 10.67837706432194, 11.71865006810371, 11.92354762278580, 12.25006588780148, 13.06962622319046, 13.63212824432603, 14.01884495690288, 14.90079689118447, 15.48378012266740
