L(s) = 1 | + 0.463·2-s + 0.981·3-s − 1.78·4-s − 3.31·5-s + 0.454·6-s − 1.75·8-s − 2.03·9-s − 1.53·10-s − 2.89·11-s − 1.75·12-s − 6.95·13-s − 3.25·15-s + 2.75·16-s + 7.66·17-s − 0.943·18-s + 1.90·19-s + 5.92·20-s − 1.34·22-s + 3.76·23-s − 1.72·24-s + 5.99·25-s − 3.22·26-s − 4.94·27-s − 3.58·29-s − 1.50·30-s + 8.42·31-s + 4.78·32-s + ⋯ |
L(s) = 1 | + 0.327·2-s + 0.566·3-s − 0.892·4-s − 1.48·5-s + 0.185·6-s − 0.620·8-s − 0.678·9-s − 0.485·10-s − 0.873·11-s − 0.505·12-s − 1.92·13-s − 0.840·15-s + 0.689·16-s + 1.86·17-s − 0.222·18-s + 0.436·19-s + 1.32·20-s − 0.286·22-s + 0.785·23-s − 0.351·24-s + 1.19·25-s − 0.631·26-s − 0.951·27-s − 0.666·29-s − 0.275·30-s + 1.51·31-s + 0.845·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8471436941\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8471436941\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 - 0.463T + 2T^{2} \) |
| 3 | \( 1 - 0.981T + 3T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 + 6.95T + 13T^{2} \) |
| 17 | \( 1 - 7.66T + 17T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 + 3.58T + 29T^{2} \) |
| 31 | \( 1 - 8.42T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 - 4.12T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 - 2.16T + 71T^{2} \) |
| 73 | \( 1 - 3.80T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 + 0.742T + 97T^{2} \) |
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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
−9.089570447753722283698765580948, −8.123208092416786016899726755679, −7.83205471079245506332622570424, −7.17810941046703915975292610178, −5.56051651273397208141599301256, −5.08376429194428681802489883026, −4.19809357637507020721546444573, −3.24108960552253890814116314602, −2.77061357522761651835471432131, −0.54568538641980600916578717235,
0.54568538641980600916578717235, 2.77061357522761651835471432131, 3.24108960552253890814116314602, 4.19809357637507020721546444573, 5.08376429194428681802489883026, 5.56051651273397208141599301256, 7.17810941046703915975292610178, 7.83205471079245506332622570424, 8.123208092416786016899726755679, 9.089570447753722283698765580948