L(s) = 1 | − 2.46·2-s + 4.08·4-s + 3.46·5-s − 7-s − 5.14·8-s − 8.55·10-s − 0.653·13-s + 2.46·14-s + 4.51·16-s + 1.13·17-s − 6.07·19-s + 14.1·20-s + 6.66·23-s + 7.01·25-s + 1.61·26-s − 4.08·28-s + 4.57·29-s + 2.79·31-s − 0.854·32-s − 2.80·34-s − 3.46·35-s − 0.439·37-s + 14.9·38-s − 17.8·40-s + 5.90·41-s + 8.70·43-s − 16.4·46-s + ⋯ |
L(s) = 1 | − 1.74·2-s + 2.04·4-s + 1.55·5-s − 0.377·7-s − 1.81·8-s − 2.70·10-s − 0.181·13-s + 0.659·14-s + 1.12·16-s + 0.275·17-s − 1.39·19-s + 3.16·20-s + 1.39·23-s + 1.40·25-s + 0.316·26-s − 0.771·28-s + 0.849·29-s + 0.502·31-s − 0.150·32-s − 0.481·34-s − 0.585·35-s − 0.0722·37-s + 2.43·38-s − 2.81·40-s + 0.921·41-s + 1.32·43-s − 2.42·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.156041921\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.156041921\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 13 | \( 1 + 0.653T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 + 6.07T + 19T^{2} \) |
| 23 | \( 1 - 6.66T + 23T^{2} \) |
| 29 | \( 1 - 4.57T + 29T^{2} \) |
| 31 | \( 1 - 2.79T + 31T^{2} \) |
| 37 | \( 1 + 0.439T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 - 0.604T + 47T^{2} \) |
| 53 | \( 1 + 9.82T + 53T^{2} \) |
| 59 | \( 1 + 1.69T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 + 6.17T + 67T^{2} \) |
| 71 | \( 1 - 5.41T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 - 2.65T + 79T^{2} \) |
| 83 | \( 1 + 6.69T + 83T^{2} \) |
| 89 | \( 1 - 0.698T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.099908736678348713686109171774, −7.22166383849783915466312319030, −6.57168645145173086323822306483, −6.18907506498326814303275533168, −5.35992946914113371590051322724, −4.37995559724002057813364698914, −2.91039452501965914964152793777, −2.42360475290526646485775332060, −1.57633295442286144064261307414, −0.72496950403575526330106149715,
0.72496950403575526330106149715, 1.57633295442286144064261307414, 2.42360475290526646485775332060, 2.91039452501965914964152793777, 4.37995559724002057813364698914, 5.35992946914113371590051322724, 6.18907506498326814303275533168, 6.57168645145173086323822306483, 7.22166383849783915466312319030, 8.099908736678348713686109171774