| L(s) = 1 | + 2-s − 1.10·3-s + 4-s + 5-s − 1.10·6-s + 4.03·7-s + 8-s − 1.78·9-s + 10-s − 0.737·11-s − 1.10·12-s − 5.77·13-s + 4.03·14-s − 1.10·15-s + 16-s + 2.90·17-s − 1.78·18-s + 7.37·19-s + 20-s − 4.43·21-s − 0.737·22-s − 1.10·24-s + 25-s − 5.77·26-s + 5.27·27-s + 4.03·28-s − 5.57·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.635·3-s + 0.5·4-s + 0.447·5-s − 0.449·6-s + 1.52·7-s + 0.353·8-s − 0.596·9-s + 0.316·10-s − 0.222·11-s − 0.317·12-s − 1.60·13-s + 1.07·14-s − 0.284·15-s + 0.250·16-s + 0.705·17-s − 0.421·18-s + 1.69·19-s + 0.223·20-s − 0.967·21-s − 0.157·22-s − 0.224·24-s + 0.200·25-s − 1.13·26-s + 1.01·27-s + 0.761·28-s − 1.03·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.068724724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.068724724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 1.10T + 3T^{2} \) |
| 7 | \( 1 - 4.03T + 7T^{2} \) |
| 11 | \( 1 + 0.737T + 11T^{2} \) |
| 13 | \( 1 + 5.77T + 13T^{2} \) |
| 17 | \( 1 - 2.90T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 + 2.10T + 41T^{2} \) |
| 43 | \( 1 - 5.93T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 2.64T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.21T + 67T^{2} \) |
| 71 | \( 1 + 1.82T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 5.44T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 7.74T + 89T^{2} \) |
| 97 | \( 1 + 9.28T + 97T^{2} \) |
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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
−7.84586595283106919928904230831, −7.52141668229699313775959697895, −6.70983361321422575635846981585, −5.50764208255105367795570297013, −5.35633830850303077406592270257, −4.92646854134748824114983269439, −3.83036950296164554250516968380, −2.75449211672014295812529853741, −2.03144273598828875450570358096, −0.896784902551901415053341125110,
0.896784902551901415053341125110, 2.03144273598828875450570358096, 2.75449211672014295812529853741, 3.83036950296164554250516968380, 4.92646854134748824114983269439, 5.35633830850303077406592270257, 5.50764208255105367795570297013, 6.70983361321422575635846981585, 7.52141668229699313775959697895, 7.84586595283106919928904230831
