| L(s) = 1 | − 2-s + 4-s + 1.60i·5-s + 1.65i·7-s − 8-s − 1.60i·10-s + (1.87 + 2.73i)11-s + 1.23i·13-s − 1.65i·14-s + 16-s − 1.74·17-s + i·19-s + 1.60i·20-s + (−1.87 − 2.73i)22-s − 5.99i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.715i·5-s + 0.625i·7-s − 0.353·8-s − 0.506i·10-s + (0.563 + 0.825i)11-s + 0.343i·13-s − 0.442i·14-s + 0.250·16-s − 0.423·17-s + 0.229i·19-s + 0.357i·20-s + (−0.398 − 0.583i)22-s − 1.24i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0163 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0163 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.402985840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402985840\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + (-1.87 - 2.73i)T \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 - 1.60iT - 5T^{2} \) |
| 7 | \( 1 - 1.65iT - 7T^{2} \) |
| 13 | \( 1 - 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 23 | \( 1 + 5.99iT - 23T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 7.98T + 37T^{2} \) |
| 41 | \( 1 + 8.60T + 41T^{2} \) |
| 43 | \( 1 + 5.30iT - 43T^{2} \) |
| 47 | \( 1 - 2.24iT - 47T^{2} \) |
| 53 | \( 1 - 4.01iT - 53T^{2} \) |
| 59 | \( 1 - 10.9iT - 59T^{2} \) |
| 61 | \( 1 + 0.355iT - 61T^{2} \) |
| 67 | \( 1 - 9.96T + 67T^{2} \) |
| 71 | \( 1 - 8.44iT - 71T^{2} \) |
| 73 | \( 1 - 1.24iT - 73T^{2} \) |
| 79 | \( 1 + 1.72iT - 79T^{2} \) |
| 83 | \( 1 - 8.04T + 83T^{2} \) |
| 89 | \( 1 + 4.10iT - 89T^{2} \) |
| 97 | \( 1 + 3.14T + 97T^{2} \) |
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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
−8.620212270785293130436412624412, −8.167195716607342972661773368454, −7.05610566117266640061331497629, −6.65905789420423989706152859853, −6.05899689156203573665916065628, −4.84483790529076287705667215532, −4.12415036203119460946580307309, −2.78666272506766606916506376319, −2.37211655044438837294615234623, −1.08153439038145249076999474908,
0.65925847946975348974680297353, 1.30704780455167519631349359702, 2.68958510030668742055065821860, 3.58440010608028372191687678253, 4.55653083859008814768608233450, 5.30728906562932161581042534220, 6.40174053736171140916674536068, 6.75443112680197751282500271427, 7.954002652594793023036410874016, 8.230634864030640758955405388889
