L(s) = 1 | − 2.44·3-s + 2.61·5-s − 0.969·7-s + 2.99·9-s − 0.473·11-s + 6.64·13-s − 6.40·15-s + 3.15·17-s − 7.48·19-s + 2.37·21-s + 7.33·23-s + 1.84·25-s + 0.0183·27-s + 0.238·29-s − 0.258·31-s + 1.16·33-s − 2.53·35-s + 5.02·37-s − 16.2·39-s − 6.95·41-s + 9.54·43-s + 7.83·45-s + 47-s − 6.05·49-s − 7.73·51-s + 11.0·53-s − 1.24·55-s + ⋯ |
L(s) = 1 | − 1.41·3-s + 1.17·5-s − 0.366·7-s + 0.997·9-s − 0.142·11-s + 1.84·13-s − 1.65·15-s + 0.766·17-s − 1.71·19-s + 0.518·21-s + 1.52·23-s + 0.369·25-s + 0.00353·27-s + 0.0443·29-s − 0.0463·31-s + 0.201·33-s − 0.429·35-s + 0.825·37-s − 2.60·39-s − 1.08·41-s + 1.45·43-s + 1.16·45-s + 0.145·47-s − 0.865·49-s − 1.08·51-s + 1.51·53-s − 0.167·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.566567565\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.566567565\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 + 0.969T + 7T^{2} \) |
| 11 | \( 1 + 0.473T + 11T^{2} \) |
| 13 | \( 1 - 6.64T + 13T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 - 0.238T + 29T^{2} \) |
| 31 | \( 1 + 0.258T + 31T^{2} \) |
| 37 | \( 1 - 5.02T + 37T^{2} \) |
| 41 | \( 1 + 6.95T + 41T^{2} \) |
| 43 | \( 1 - 9.54T + 43T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 0.263T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 - 5.43T + 67T^{2} \) |
| 71 | \( 1 - 4.61T + 71T^{2} \) |
| 73 | \( 1 + 1.81T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 9.15T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.184730947566790752349912165221, −6.91907874403258240758587898867, −6.50794734352908634838711745569, −5.82353903448015044050105081247, −5.58905614559226379294582904989, −4.63947681020276776957947149441, −3.76900031741041765015433936321, −2.70566730558369447745971291573, −1.57223668269623132539086526121, −0.75604230355958274306682565926,
0.75604230355958274306682565926, 1.57223668269623132539086526121, 2.70566730558369447745971291573, 3.76900031741041765015433936321, 4.63947681020276776957947149441, 5.58905614559226379294582904989, 5.82353903448015044050105081247, 6.50794734352908634838711745569, 6.91907874403258240758587898867, 8.184730947566790752349912165221