| 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 − 2·29-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 − 0.371·29-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 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.19820898032230, −14.67047428058126, −14.19894794067098, −13.91320237663653, −13.00795995458803, −12.35756968381154, −12.09926016174339, −11.42720413283072, −10.94782598072555, −10.37212210742108, −9.659080824579742, −9.401663165656335, −8.610571703917027, −8.280549876219665, −7.783807836975627, −7.200570976432945, −6.498251112292851, −6.128655264785496, −5.211426881694055, −4.421977226860003, −3.791958701405661, −3.411900511072232, −2.321359954989814, −1.833553054115496, −1.062990064051637, 0,
1.062990064051637, 1.833553054115496, 2.321359954989814, 3.411900511072232, 3.791958701405661, 4.421977226860003, 5.211426881694055, 6.128655264785496, 6.498251112292851, 7.200570976432945, 7.783807836975627, 8.280549876219665, 8.610571703917027, 9.401663165656335, 9.659080824579742, 10.37212210742108, 10.94782598072555, 11.42720413283072, 12.09926016174339, 12.35756968381154, 13.00795995458803, 13.91320237663653, 14.19894794067098, 14.67047428058126, 15.19820898032230
