L(s) = 1 | − 3.46·5-s + 4.89·7-s + 5.65·11-s + 6.99·25-s − 10.3·29-s + 4.89·31-s − 16.9·35-s + 16.9·49-s + 3.46·53-s − 19.5·55-s + 11.3·59-s + 14·73-s + 27.7·77-s − 14.6·79-s − 5.65·83-s + 2·97-s + 3.46·101-s + 14.6·103-s + 11.3·107-s + ⋯ |
L(s) = 1 | − 1.54·5-s + 1.85·7-s + 1.70·11-s + 1.39·25-s − 1.92·29-s + 0.879·31-s − 2.86·35-s + 2.42·49-s + 0.475·53-s − 2.64·55-s + 1.47·59-s + 1.63·73-s + 3.15·77-s − 1.65·79-s − 0.620·83-s + 0.203·97-s + 0.344·101-s + 1.44·103-s + 1.09·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.839692273\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.839692273\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 4.89T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.675309294489954983821477923112, −8.315743272799727691590526193401, −7.47840018339691696960521780702, −7.01010865517931442907142754793, −5.79646602759485350974595448665, −4.76938310557008972052183149330, −4.15588544064717398883299464868, −3.56534460915115997821247876182, −1.97306422197523117783472654739, −0.938690730154911277958600849100,
0.938690730154911277958600849100, 1.97306422197523117783472654739, 3.56534460915115997821247876182, 4.15588544064717398883299464868, 4.76938310557008972052183149330, 5.79646602759485350974595448665, 7.01010865517931442907142754793, 7.47840018339691696960521780702, 8.315743272799727691590526193401, 8.675309294489954983821477923112