| L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s + 7-s + 8-s + 9-s + 4·10-s + 11-s + 12-s + 14-s + 4·15-s + 16-s − 2·17-s + 18-s + 19-s + 4·20-s + 21-s + 22-s + 5·23-s + 24-s + 11·25-s + 27-s + 28-s + 2·29-s + 4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s + 0.288·12-s + 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s + 0.894·20-s + 0.218·21-s + 0.213·22-s + 1.04·23-s + 0.204·24-s + 11/5·25-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70602 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70602 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.58083880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.58083880\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.92083967296622, −13.69031549530864, −13.42118897562446, −12.72944925873778, −12.36457345471098, −11.70805021890996, −11.03027078226836, −10.61394728620067, −10.06780720280610, −9.582600059691099, −9.095041547774012, −8.619010999434929, −8.014981376334822, −7.233118810804160, −6.760244398516029, −6.255837197497919, −5.780420065914911, −5.116753586771859, −4.694435076124919, −4.103530235536144, −3.186917508072832, −2.645640945567922, −2.267217357649157, −1.449630626028994, −1.000412225948248,
1.000412225948248, 1.449630626028994, 2.267217357649157, 2.645640945567922, 3.186917508072832, 4.103530235536144, 4.694435076124919, 5.116753586771859, 5.780420065914911, 6.255837197497919, 6.760244398516029, 7.233118810804160, 8.014981376334822, 8.619010999434929, 9.095041547774012, 9.582600059691099, 10.06780720280610, 10.61394728620067, 11.03027078226836, 11.70805021890996, 12.36457345471098, 12.72944925873778, 13.42118897562446, 13.69031549530864, 13.92083967296622
