| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 1.89·7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 2.77·13-s + 1.89·14-s − 15-s + 16-s − 17-s + 18-s + 2.77·19-s + 20-s − 1.89·21-s + 22-s + 4.37·23-s − 24-s + 25-s + 2.77·26-s − 27-s + 1.89·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.716·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.770·13-s + 0.506·14-s − 0.258·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.636·19-s + 0.223·20-s − 0.413·21-s + 0.213·22-s + 0.911·23-s − 0.204·24-s + 0.200·25-s + 0.544·26-s − 0.192·27-s + 0.358·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.587676293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.587676293\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 1.89T + 7T^{2} \) |
| 13 | \( 1 - 2.77T + 13T^{2} \) |
| 19 | \( 1 - 2.77T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 + 8.88T + 31T^{2} \) |
| 37 | \( 1 - 3.73T + 37T^{2} \) |
| 41 | \( 1 + 0.476T + 41T^{2} \) |
| 43 | \( 1 - 6.34T + 43T^{2} \) |
| 47 | \( 1 + 0.476T + 47T^{2} \) |
| 53 | \( 1 - 4.75T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 8.21T + 61T^{2} \) |
| 67 | \( 1 + 9.08T + 67T^{2} \) |
| 71 | \( 1 + 2.51T + 71T^{2} \) |
| 73 | \( 1 + 2.75T + 73T^{2} \) |
| 79 | \( 1 - 8.99T + 79T^{2} \) |
| 83 | \( 1 - 6.81T + 83T^{2} \) |
| 89 | \( 1 - 3.76T + 89T^{2} \) |
| 97 | \( 1 - 3.33T + 97T^{2} \) |
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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
−7.967186676708707631683242193023, −7.26466619715841346889327833984, −6.54271974481205804066823400035, −5.89220052821573856402395983484, −5.24856979149036929962553106611, −4.62744993404221994045898959828, −3.82042785257619818138517631902, −2.90208407990799584368107985133, −1.81303500758322067112998466214, −1.01487324782286991563178019515,
1.01487324782286991563178019515, 1.81303500758322067112998466214, 2.90208407990799584368107985133, 3.82042785257619818138517631902, 4.62744993404221994045898959828, 5.24856979149036929962553106611, 5.89220052821573856402395983484, 6.54271974481205804066823400035, 7.26466619715841346889327833984, 7.967186676708707631683242193023
