L(s) = 1 | − 1.74·2-s + 2.61·3-s + 1.04·4-s − 0.0450·5-s − 4.57·6-s − 1.64·7-s + 1.65·8-s + 3.85·9-s + 0.0786·10-s − 1.43·11-s + 2.74·12-s + 2.18·13-s + 2.87·14-s − 0.117·15-s − 4.99·16-s − 4.11·17-s − 6.74·18-s − 7.92·19-s − 0.0472·20-s − 4.31·21-s + 2.50·22-s + 8.12·23-s + 4.34·24-s − 4.99·25-s − 3.81·26-s + 2.25·27-s − 1.72·28-s + ⋯ |
L(s) = 1 | − 1.23·2-s + 1.51·3-s + 0.524·4-s − 0.0201·5-s − 1.86·6-s − 0.622·7-s + 0.586·8-s + 1.28·9-s + 0.0248·10-s − 0.432·11-s + 0.793·12-s + 0.606·13-s + 0.768·14-s − 0.0304·15-s − 1.24·16-s − 0.997·17-s − 1.58·18-s − 1.81·19-s − 0.0105·20-s − 0.940·21-s + 0.534·22-s + 1.69·23-s + 0.887·24-s − 0.999·25-s − 0.748·26-s + 0.433·27-s − 0.326·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 0.0450T + 5T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 23 | \( 1 - 8.12T + 23T^{2} \) |
| 29 | \( 1 - 5.25T + 29T^{2} \) |
| 31 | \( 1 - 8.11T + 31T^{2} \) |
| 41 | \( 1 + 3.25T + 41T^{2} \) |
| 43 | \( 1 + 1.02T + 43T^{2} \) |
| 47 | \( 1 - 0.633T + 47T^{2} \) |
| 53 | \( 1 + 8.79T + 53T^{2} \) |
| 59 | \( 1 + 7.03T + 59T^{2} \) |
| 61 | \( 1 + 7.72T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 9.59T + 73T^{2} \) |
| 79 | \( 1 - 9.98T + 79T^{2} \) |
| 83 | \( 1 + 1.30T + 83T^{2} \) |
| 89 | \( 1 + 7.22T + 89T^{2} \) |
| 97 | \( 1 + 9.33T + 97T^{2} \) |
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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
−8.311290440140722621216408139561, −7.79969433242761233894514804246, −6.77529906139400643409884244061, −6.39033590746461200003978607610, −4.78676725240838316305106022779, −4.13122585853297041487441448204, −3.08106502961562834405875638555, −2.39530570161119648976127013730, −1.46095661874291438979855073383, 0,
1.46095661874291438979855073383, 2.39530570161119648976127013730, 3.08106502961562834405875638555, 4.13122585853297041487441448204, 4.78676725240838316305106022779, 6.39033590746461200003978607610, 6.77529906139400643409884244061, 7.79969433242761233894514804246, 8.311290440140722621216408139561