L(s) = 1 | − 2-s − 3-s + 4-s + 3.64·5-s + 6-s − 8-s + 9-s − 3.64·10-s + 0.645·11-s − 12-s − 13-s − 3.64·15-s + 16-s + 6.64·17-s − 18-s − 5·19-s + 3.64·20-s − 0.645·22-s − 2.35·23-s + 24-s + 8.29·25-s + 26-s − 27-s + 4.29·29-s + 3.64·30-s − 3.29·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.63·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.15·10-s + 0.194·11-s − 0.288·12-s − 0.277·13-s − 0.941·15-s + 0.250·16-s + 1.61·17-s − 0.235·18-s − 1.14·19-s + 0.815·20-s − 0.137·22-s − 0.490·23-s + 0.204·24-s + 1.65·25-s + 0.196·26-s − 0.192·27-s + 0.796·29-s + 0.665·30-s − 0.591·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.610985097\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.610985097\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 3.64T + 5T^{2} \) |
| 11 | \( 1 - 0.645T + 11T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 + 3.29T + 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 - 2.35T + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 7.58T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 0.937T + 97T^{2} \) |
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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
−8.510113237566213736411873754387, −7.88104163522829595774433616819, −6.78139031334244418283531266880, −6.38694771981900348351044676853, −5.59766974614110183227624729728, −5.09070356934360039320653836404, −3.82222687670851610781373462422, −2.59368521918941962983976828447, −1.83280741425929237456236240185, −0.871994862396183004263668419787,
0.871994862396183004263668419787, 1.83280741425929237456236240185, 2.59368521918941962983976828447, 3.82222687670851610781373462422, 5.09070356934360039320653836404, 5.59766974614110183227624729728, 6.38694771981900348351044676853, 6.78139031334244418283531266880, 7.88104163522829595774433616819, 8.510113237566213736411873754387