L(s) = 1 | − 1.93·3-s + 5-s + 2.38·7-s + 0.735·9-s − 5.33·11-s + 4.53·13-s − 1.93·15-s + 1.81·17-s − 7.00·19-s − 4.60·21-s − 23-s + 25-s + 4.37·27-s + 0.118·29-s − 0.884·31-s + 10.3·33-s + 2.38·35-s − 7.51·37-s − 8.77·39-s − 1.45·41-s + 0.735·45-s + 10.4·47-s − 1.32·49-s − 3.50·51-s + 9.42·53-s − 5.33·55-s + 13.5·57-s + ⋯ |
L(s) = 1 | − 1.11·3-s + 0.447·5-s + 0.900·7-s + 0.245·9-s − 1.60·11-s + 1.25·13-s − 0.499·15-s + 0.440·17-s − 1.60·19-s − 1.00·21-s − 0.208·23-s + 0.200·25-s + 0.842·27-s + 0.0219·29-s − 0.158·31-s + 1.79·33-s + 0.402·35-s − 1.23·37-s − 1.40·39-s − 0.226·41-s + 0.109·45-s + 1.52·47-s − 0.189·49-s − 0.491·51-s + 1.29·53-s − 0.719·55-s + 1.79·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.195870067\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.195870067\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 + 5.33T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 29 | \( 1 - 0.118T + 29T^{2} \) |
| 31 | \( 1 + 0.884T + 31T^{2} \) |
| 37 | \( 1 + 7.51T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 9.42T + 53T^{2} \) |
| 59 | \( 1 + 7.79T + 59T^{2} \) |
| 61 | \( 1 - 2.80T + 61T^{2} \) |
| 67 | \( 1 - 3.11T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 - 2.89T + 89T^{2} \) |
| 97 | \( 1 + 1.97T + 97T^{2} \) |
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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
−8.023699736514879041326447714515, −7.10533846565421788209415221300, −6.29072067959825407158641953276, −5.75297096242678375820701916330, −5.21989160436181316264032534969, −4.62166241842664810479992192004, −3.65812300046115252611864907637, −2.52615600616078501121701866632, −1.72310045256650954653785043803, −0.58289239036861643127728367958,
0.58289239036861643127728367958, 1.72310045256650954653785043803, 2.52615600616078501121701866632, 3.65812300046115252611864907637, 4.62166241842664810479992192004, 5.21989160436181316264032534969, 5.75297096242678375820701916330, 6.29072067959825407158641953276, 7.10533846565421788209415221300, 8.023699736514879041326447714515