L(s) = 1 | − 1.81·2-s − 1.25·3-s + 1.30·4-s − 0.160·5-s + 2.28·6-s + 4.31·7-s + 1.25·8-s − 1.42·9-s + 0.291·10-s − 0.148·11-s − 1.64·12-s + 1.68·13-s − 7.85·14-s + 0.201·15-s − 4.90·16-s − 5.73·17-s + 2.58·18-s + 4.22·19-s − 0.210·20-s − 5.42·21-s + 0.269·22-s − 3.75·23-s − 1.57·24-s − 4.97·25-s − 3.06·26-s + 5.55·27-s + 5.65·28-s + ⋯ |
L(s) = 1 | − 1.28·2-s − 0.725·3-s + 0.654·4-s − 0.0717·5-s + 0.932·6-s + 1.63·7-s + 0.443·8-s − 0.474·9-s + 0.0922·10-s − 0.0447·11-s − 0.474·12-s + 0.467·13-s − 2.09·14-s + 0.0520·15-s − 1.22·16-s − 1.39·17-s + 0.610·18-s + 0.970·19-s − 0.0469·20-s − 1.18·21-s + 0.0575·22-s − 0.782·23-s − 0.321·24-s − 0.994·25-s − 0.601·26-s + 1.06·27-s + 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 3 | \( 1 + 1.25T + 3T^{2} \) |
| 5 | \( 1 + 0.160T + 5T^{2} \) |
| 7 | \( 1 - 4.31T + 7T^{2} \) |
| 11 | \( 1 + 0.148T + 11T^{2} \) |
| 13 | \( 1 - 1.68T + 13T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 23 | \( 1 + 3.75T + 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 - 8.64T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 - 1.98T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 1.42T + 67T^{2} \) |
| 71 | \( 1 - 7.56T + 71T^{2} \) |
| 73 | \( 1 - 4.99T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 6.11T + 83T^{2} \) |
| 89 | \( 1 - 8.33T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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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
−8.660000833788523673408286295975, −8.266846752439302018292357317386, −7.55280417311951547081242580011, −6.66919183095750197863815175669, −5.60823757249561454995325135098, −4.90348580425042509243342455655, −3.99900206044438133759396830129, −2.23944654455986462378402611645, −1.34509169526705467464585047458, 0,
1.34509169526705467464585047458, 2.23944654455986462378402611645, 3.99900206044438133759396830129, 4.90348580425042509243342455655, 5.60823757249561454995325135098, 6.66919183095750197863815175669, 7.55280417311951547081242580011, 8.266846752439302018292357317386, 8.660000833788523673408286295975