L(s) = 1 | + 0.339·3-s + 3.88i·7-s − 2.88·9-s − 1.54i·11-s + (−3.54 + 0.660i)13-s − 2.86i·19-s + 1.32i·21-s − 5.42·23-s − 2·27-s − 5.20·29-s − 6.22i·31-s − 0.524i·33-s − 8.56i·37-s + (−1.20 + 0.224i)39-s + 9.08i·41-s + ⋯ |
L(s) = 1 | + 0.196·3-s + 1.46i·7-s − 0.961·9-s − 0.465i·11-s + (−0.983 + 0.183i)13-s − 0.657i·19-s + 0.288i·21-s − 1.13·23-s − 0.384·27-s − 0.966·29-s − 1.11i·31-s − 0.0913i·33-s − 1.40i·37-s + (−0.192 + 0.0359i)39-s + 1.41i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1374167201\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1374167201\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.54 - 0.660i)T \) |
good | 3 | \( 1 - 0.339T + 3T^{2} \) |
| 7 | \( 1 - 3.88iT - 7T^{2} \) |
| 11 | \( 1 + 1.54iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.86iT - 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 + 6.22iT - 31T^{2} \) |
| 37 | \( 1 + 8.56iT - 37T^{2} \) |
| 41 | \( 1 - 9.08iT - 41T^{2} \) |
| 43 | \( 1 + 0.980T + 43T^{2} \) |
| 47 | \( 1 + 6.52iT - 47T^{2} \) |
| 53 | \( 1 + 6.44T + 53T^{2} \) |
| 59 | \( 1 + 4.45iT - 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 6.97iT - 67T^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + 3.43iT - 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 8.56iT - 83T^{2} \) |
| 89 | \( 1 - 17.1iT - 89T^{2} \) |
| 97 | \( 1 - 13.7iT - 97T^{2} \) |
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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
−9.743565672824767220579098996756, −9.306508133523943549390575004682, −8.460256380767631944193918897998, −7.890195188519750149326484046361, −6.71433804695138246085048721181, −5.71285632083758719046122859681, −5.34027007894149161674392779665, −3.99936701222558328837327884076, −2.74803429033842967252080107060, −2.18940107845370987106781579628,
0.05082300769179275978635054574, 1.72921831748911871896396452391, 3.04986317130336753196967359368, 3.97402813348519013816276408939, 4.88135467252149959686939543461, 5.88552311593861941103984475847, 6.92327299160826372406129273977, 7.61391600342673746304201585630, 8.255091635763793527324183269451, 9.323565462871590172063017649417