L(s) = 1 | − 2.12i·3-s + (1.80 − 1.32i)5-s − 4.12i·7-s − 1.51·9-s + 2.64·11-s − 2.51i·13-s + (−2.80 − 3.83i)15-s − 0.515i·17-s + 19-s − 8.76·21-s + 3.09i·23-s + (1.51 − 4.76i)25-s − 3.15i·27-s + 7.79·29-s − 3.67·31-s + ⋯ |
L(s) = 1 | − 1.22i·3-s + (0.807 − 0.590i)5-s − 1.55i·7-s − 0.505·9-s + 0.795·11-s − 0.697i·13-s + (−0.724 − 0.990i)15-s − 0.124i·17-s + 0.229·19-s − 1.91·21-s + 0.645i·23-s + (0.303 − 0.952i)25-s − 0.607i·27-s + 1.44·29-s − 0.659·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.126918900\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.126918900\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.80 + 1.32i)T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 2.12iT - 3T^{2} \) |
| 7 | \( 1 + 4.12iT - 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 2.51iT - 13T^{2} \) |
| 17 | \( 1 + 0.515iT - 17T^{2} \) |
| 23 | \( 1 - 3.09iT - 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 + 3.67T + 31T^{2} \) |
| 37 | \( 1 - 10.2iT - 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 - 8.64iT - 43T^{2} \) |
| 47 | \( 1 - 4.96iT - 47T^{2} \) |
| 53 | \( 1 + 5.48iT - 53T^{2} \) |
| 59 | \( 1 - 3.15T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 7.40iT - 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 2.70iT - 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 - 3.28iT - 83T^{2} \) |
| 89 | \( 1 - 7.60T + 89T^{2} \) |
| 97 | \( 1 - 3.93iT - 97T^{2} \) |
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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
−9.171933990483687966237148582674, −8.057085497460511414677440309349, −7.59455553992395493022467630754, −6.64415984351016417852237465371, −6.21888796602290388551054681229, −4.99857090130780821303065609513, −4.10329198321817459663185531456, −2.82841014470188987574386970386, −1.37700099770410846003944851831, −0.966522311583551460657060467510,
1.90531732184374591245567099455, 2.81563292049774494569365724622, 3.86768833534924003943797860696, 4.81995726380412686457535447543, 5.69126541111945128749724183754, 6.28375296144453255367345187573, 7.24896796430046224289091782950, 8.699874178461011695482966190323, 9.150017554132917392884315724977, 9.583915465619477256569090429046