| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 4.24·7-s − 8-s + 9-s − 10-s − 0.801·11-s − 12-s − 4.24·14-s − 15-s + 16-s + 7.43·17-s − 18-s + 3.74·19-s + 20-s − 4.24·21-s + 0.801·22-s − 2.54·23-s + 24-s + 25-s − 27-s + 4.24·28-s + 5.63·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.60·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.241·11-s − 0.288·12-s − 1.13·14-s − 0.258·15-s + 0.250·16-s + 1.80·17-s − 0.235·18-s + 0.858·19-s + 0.223·20-s − 0.926·21-s + 0.170·22-s − 0.530·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.802·28-s + 1.04·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.770949286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.770949286\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 0.801T + 11T^{2} \) |
| 17 | \( 1 - 7.43T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 + 2.54T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 + 1.69T + 31T^{2} \) |
| 37 | \( 1 + 9.67T + 37T^{2} \) |
| 41 | \( 1 - 8.85T + 41T^{2} \) |
| 43 | \( 1 + 6.29T + 43T^{2} \) |
| 47 | \( 1 - 7.53T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 9.58T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 7.82T + 67T^{2} \) |
| 71 | \( 1 + 2.04T + 71T^{2} \) |
| 73 | \( 1 - 2.17T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 3.61T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 - 6.07T + 97T^{2} \) |
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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
−8.256338810108619233395203345501, −7.50114394396329682894789246800, −7.09504654172793873878816244274, −5.85337484722427987022647775922, −5.47966051912567994504883850190, −4.79931493252929415400373701477, −3.72181912380671196159341034477, −2.57440981653289131985948537652, −1.56738241539015144873789532707, −0.927535903782753781866149163498,
0.927535903782753781866149163498, 1.56738241539015144873789532707, 2.57440981653289131985948537652, 3.72181912380671196159341034477, 4.79931493252929415400373701477, 5.47966051912567994504883850190, 5.85337484722427987022647775922, 7.09504654172793873878816244274, 7.50114394396329682894789246800, 8.256338810108619233395203345501
