L(s) = 1 | + 0.445·2-s + 3-s − 1.80·4-s − 5-s + 0.445·6-s + 7-s − 1.69·8-s + 9-s − 0.445·10-s − 0.445·11-s − 1.80·12-s + 0.445·14-s − 15-s + 2.85·16-s − 0.753·17-s + 0.445·18-s + 3.40·19-s + 1.80·20-s + 21-s − 0.198·22-s − 2.64·23-s − 1.69·24-s − 4·25-s + 27-s − 1.80·28-s − 2.13·29-s − 0.445·30-s + ⋯ |
L(s) = 1 | + 0.314·2-s + 0.577·3-s − 0.900·4-s − 0.447·5-s + 0.181·6-s + 0.377·7-s − 0.598·8-s + 0.333·9-s − 0.140·10-s − 0.134·11-s − 0.520·12-s + 0.118·14-s − 0.258·15-s + 0.712·16-s − 0.182·17-s + 0.104·18-s + 0.781·19-s + 0.402·20-s + 0.218·21-s − 0.0422·22-s − 0.551·23-s − 0.345·24-s − 0.800·25-s + 0.192·27-s − 0.340·28-s − 0.396·29-s − 0.0812·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.445T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 0.445T + 11T^{2} \) |
| 17 | \( 1 + 0.753T + 17T^{2} \) |
| 19 | \( 1 - 3.40T + 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 + 2.13T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 + 8.20T + 37T^{2} \) |
| 41 | \( 1 + 1.51T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 - 4.16T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 + 0.740T + 67T^{2} \) |
| 71 | \( 1 + 1.66T + 71T^{2} \) |
| 73 | \( 1 + 5.74T + 73T^{2} \) |
| 79 | \( 1 - 0.719T + 79T^{2} \) |
| 83 | \( 1 - 0.131T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.290869247728979589450594560781, −7.61290675687922859644606532921, −6.84871285854619955189631565162, −5.64885582003639708585284987656, −5.14758981996351563025383767815, −4.11538371912158495722877716128, −3.72298479040942644822523626151, −2.71432169642586701753418947213, −1.48268808518074553761528145771, 0,
1.48268808518074553761528145771, 2.71432169642586701753418947213, 3.72298479040942644822523626151, 4.11538371912158495722877716128, 5.14758981996351563025383767815, 5.64885582003639708585284987656, 6.84871285854619955189631565162, 7.61290675687922859644606532921, 8.290869247728979589450594560781