L(s) = 1 | + 2.60·2-s + 3-s + 4.78·4-s + 1.51·5-s + 2.60·6-s + 7-s + 7.26·8-s + 9-s + 3.94·10-s + 0.949·11-s + 4.78·12-s + 2.60·14-s + 1.51·15-s + 9.34·16-s + 5.20·17-s + 2.60·18-s − 5.53·19-s + 7.24·20-s + 21-s + 2.47·22-s − 3.94·23-s + 7.26·24-s − 2.70·25-s + 27-s + 4.78·28-s + 5.11·29-s + 3.94·30-s + ⋯ |
L(s) = 1 | + 1.84·2-s + 0.577·3-s + 2.39·4-s + 0.676·5-s + 1.06·6-s + 0.377·7-s + 2.56·8-s + 0.333·9-s + 1.24·10-s + 0.286·11-s + 1.38·12-s + 0.696·14-s + 0.390·15-s + 2.33·16-s + 1.26·17-s + 0.614·18-s − 1.26·19-s + 1.62·20-s + 0.218·21-s + 0.527·22-s − 0.823·23-s + 1.48·24-s − 0.541·25-s + 0.192·27-s + 0.904·28-s + 0.949·29-s + 0.719·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.940453624\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.940453624\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 5 | \( 1 - 1.51T + 5T^{2} \) |
| 11 | \( 1 - 0.949T + 11T^{2} \) |
| 17 | \( 1 - 5.20T + 17T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 - 5.11T + 29T^{2} \) |
| 31 | \( 1 + 9.26T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 6.97T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 6.64T + 59T^{2} \) |
| 61 | \( 1 + 3.07T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 7.77T + 71T^{2} \) |
| 73 | \( 1 - 8.64T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 8.13T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.341425121966441341110495727148, −7.63656080980308389195453991365, −6.78574275619007788568794472119, −6.12181701242775105076856535868, −5.45646457097459462024572330190, −4.73631997801981413058507023338, −3.85770577931358111188839953987, −3.30012888716408539761939667397, −2.18751569062277776753407496042, −1.68121519757464093455993869520,
1.68121519757464093455993869520, 2.18751569062277776753407496042, 3.30012888716408539761939667397, 3.85770577931358111188839953987, 4.73631997801981413058507023338, 5.45646457097459462024572330190, 6.12181701242775105076856535868, 6.78574275619007788568794472119, 7.63656080980308389195453991365, 8.341425121966441341110495727148