L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 5.46·11-s − 12-s − 3.46·13-s − 14-s + 15-s + 16-s − 2·17-s + 18-s + 4·19-s − 20-s + 21-s − 5.46·22-s − 23-s − 24-s + 25-s − 3.46·26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.64·11-s − 0.288·12-s − 0.960·13-s − 0.267·14-s + 0.258·15-s + 0.250·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 1.16·22-s − 0.208·23-s − 0.204·24-s + 0.200·25-s − 0.679·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.433400739\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.433400739\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 7.46T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 5.46T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 5.46T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 0.535T + 97T^{2} \) |
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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
−7.78913333773056813669322689927, −7.61059373768881025404973658949, −6.81155214602465211206634993747, −5.88025985741817719744829770923, −5.32380053927102334118857648378, −4.69112229139409102235617198142, −3.87906868108602410022842479835, −2.87721970371921524626372772505, −2.22351188786777668580105570770, −0.57714033020378025414858070421,
0.57714033020378025414858070421, 2.22351188786777668580105570770, 2.87721970371921524626372772505, 3.87906868108602410022842479835, 4.69112229139409102235617198142, 5.32380053927102334118857648378, 5.88025985741817719744829770923, 6.81155214602465211206634993747, 7.61059373768881025404973658949, 7.78913333773056813669322689927