| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 1.84·7-s + 8-s + 9-s + 10-s + 0.832·11-s + 12-s + 3.21·13-s + 1.84·14-s + 15-s + 16-s − 7.95·17-s + 18-s + 4.20·19-s + 20-s + 1.84·21-s + 0.832·22-s − 6.57·23-s + 24-s + 25-s + 3.21·26-s + 27-s + 1.84·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.697·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.251·11-s + 0.288·12-s + 0.892·13-s + 0.493·14-s + 0.258·15-s + 0.250·16-s − 1.92·17-s + 0.235·18-s + 0.965·19-s + 0.223·20-s + 0.402·21-s + 0.177·22-s − 1.37·23-s + 0.204·24-s + 0.200·25-s + 0.631·26-s + 0.192·27-s + 0.348·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.580442974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.580442974\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 - 0.832T + 11T^{2} \) |
| 13 | \( 1 - 3.21T + 13T^{2} \) |
| 17 | \( 1 + 7.95T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 + 6.57T + 23T^{2} \) |
| 29 | \( 1 - 3.03T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 + 5.03T + 43T^{2} \) |
| 47 | \( 1 - 9.11T + 47T^{2} \) |
| 53 | \( 1 - 3.95T + 53T^{2} \) |
| 59 | \( 1 + 3.69T + 59T^{2} \) |
| 61 | \( 1 - 5.40T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 3.55T + 83T^{2} \) |
| 89 | \( 1 + 6.55T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−9.841826683452705354474916719248, −8.903475222779944931848411012539, −8.272519577689492539295003287103, −7.25269864651798043706755608691, −6.41898449591979531711839101253, −5.53836063727897053152438324123, −4.49783999740777161847911303981, −3.75491063720937523789328863066, −2.49119996496987885483086559808, −1.57488951409532402202085054848,
1.57488951409532402202085054848, 2.49119996496987885483086559808, 3.75491063720937523789328863066, 4.49783999740777161847911303981, 5.53836063727897053152438324123, 6.41898449591979531711839101253, 7.25269864651798043706755608691, 8.272519577689492539295003287103, 8.903475222779944931848411012539, 9.841826683452705354474916719248
