| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 1.06·7-s − 8-s + 9-s + 10-s − 3.48·11-s − 12-s − 3.47·13-s − 1.06·14-s + 15-s + 16-s − 18-s − 7.34·19-s − 20-s − 1.06·21-s + 3.48·22-s − 8.67·23-s + 24-s + 25-s + 3.47·26-s − 27-s + 1.06·28-s − 2.62·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.401·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.05·11-s − 0.288·12-s − 0.962·13-s − 0.283·14-s + 0.258·15-s + 0.250·16-s − 0.235·18-s − 1.68·19-s − 0.223·20-s − 0.231·21-s + 0.743·22-s − 1.80·23-s + 0.204·24-s + 0.200·25-s + 0.680·26-s − 0.192·27-s + 0.200·28-s − 0.487·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08298397891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08298397891\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 1.06T + 7T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 + 8.67T + 23T^{2} \) |
| 29 | \( 1 + 2.62T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 5.53T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 7.02T + 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 - 9.86T + 53T^{2} \) |
| 59 | \( 1 + 6.17T + 59T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 9.60T + 73T^{2} \) |
| 79 | \( 1 + 4.35T + 79T^{2} \) |
| 83 | \( 1 + 9.38T + 83T^{2} \) |
| 89 | \( 1 - 5.59T + 89T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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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
−7.63049558401568612245015736842, −7.35449306356568841247660671567, −6.48717641041501488409341959881, −5.72309913584331833858972139882, −5.09175917981715536300772921577, −4.30671613461966053825528624276, −3.50298787099139986729099620924, −2.26422441559639657219183354746, −1.82340303323534953617722194287, −0.15092294961231145513515682943,
0.15092294961231145513515682943, 1.82340303323534953617722194287, 2.26422441559639657219183354746, 3.50298787099139986729099620924, 4.30671613461966053825528624276, 5.09175917981715536300772921577, 5.72309913584331833858972139882, 6.48717641041501488409341959881, 7.35449306356568841247660671567, 7.63049558401568612245015736842
