| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s + 7-s − 3·8-s + 9-s − 10-s − 4·11-s − 12-s + 14-s − 15-s − 16-s + 17-s + 18-s + 4·19-s + 20-s + 21-s − 4·22-s − 3·24-s + 25-s + 27-s − 28-s + 6·29-s − 30-s + 8·31-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 − 1.06·8-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 + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.852·22-s − 0.612·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.434367171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.434367171\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.77473178380638, −12.21123498651659, −11.99277397137264, −11.49163306202942, −10.80039342505830, −10.37217037412602, −10.00590008112022, −9.309590504283490, −9.059414858241172, −8.456471128812571, −7.959392777064710, −7.695878952210563, −7.299788194169808, −6.361305319106481, −6.080287846177586, −5.465139637866593, −4.862232037892726, −4.540362079679095, −4.261949327936828, −3.343781360128421, −2.981250308030333, −2.764749572219453, −1.873875249618698, −1.038865596447432, −0.4791164776268168,
0.4791164776268168, 1.038865596447432, 1.873875249618698, 2.764749572219453, 2.981250308030333, 3.343781360128421, 4.261949327936828, 4.540362079679095, 4.862232037892726, 5.465139637866593, 6.080287846177586, 6.361305319106481, 7.299788194169808, 7.695878952210563, 7.959392777064710, 8.456471128812571, 9.059414858241172, 9.309590504283490, 10.00590008112022, 10.37217037412602, 10.80039342505830, 11.49163306202942, 11.99277397137264, 12.21123498651659, 12.77473178380638
