L(s) = 1 | − 3-s − 5-s − 2.60i·7-s + 9-s + 1.74i·11-s − 2.60i·13-s + 15-s − 0.802·17-s + (0.197 + 4.35i)19-s + 2.60i·21-s − 1.23i·23-s + 25-s − 27-s + 1.74i·29-s + 8.15·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.985i·7-s + 0.333·9-s + 0.526i·11-s − 0.723i·13-s + 0.258·15-s − 0.194·17-s + (0.0454 + 0.998i)19-s + 0.569i·21-s − 0.256i·23-s + 0.200·25-s − 0.192·27-s + 0.324i·29-s + 1.46·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0454 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0454 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.077512382\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.077512382\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (-0.197 - 4.35i)T \) |
good | 7 | \( 1 + 2.60iT - 7T^{2} \) |
| 11 | \( 1 - 1.74iT - 11T^{2} \) |
| 13 | \( 1 + 2.60iT - 13T^{2} \) |
| 17 | \( 1 + 0.802T + 17T^{2} \) |
| 23 | \( 1 + 1.23iT - 23T^{2} \) |
| 29 | \( 1 - 1.74iT - 29T^{2} \) |
| 31 | \( 1 - 8.15T + 31T^{2} \) |
| 37 | \( 1 + 2.97iT - 37T^{2} \) |
| 41 | \( 1 + 6.96iT - 41T^{2} \) |
| 43 | \( 1 + 2.97iT - 43T^{2} \) |
| 47 | \( 1 - 9.93iT - 47T^{2} \) |
| 53 | \( 1 - 1.23iT - 53T^{2} \) |
| 59 | \( 1 - 1.60T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 9.35T + 67T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 - 4.72iT - 83T^{2} \) |
| 89 | \( 1 + 12.5iT - 89T^{2} \) |
| 97 | \( 1 + 8.85iT - 97T^{2} \) |
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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
−7.956036394535218266590007544820, −7.43563485698521933764535788444, −6.77439042233843876745341073606, −5.98195070349547571245932593884, −5.16757008533308605626568085436, −4.32795370068358497285731687450, −3.81107837955976643731272972020, −2.73542969040162127143386050123, −1.42754528244400970001071374784, −0.42004704081597296589588180401,
0.926937810301434658420741285502, 2.23215539739749270541178352634, 3.05746133603291805649396212742, 4.12434096134390871778137292017, 4.86110298234246375981173132607, 5.53776747292727632558226498954, 6.43042844497445137694550016029, 6.81035091562320710248921562521, 7.87196483521485702807652076299, 8.501131373961558705722737304224