| L(s) = 1 | − 2-s − 0.00489·3-s + 4-s − 4.12·5-s + 0.00489·6-s − 1.34·7-s − 8-s − 2.99·9-s + 4.12·10-s + 1.51·11-s − 0.00489·12-s + 3.47·13-s + 1.34·14-s + 0.0201·15-s + 16-s + 0.907·17-s + 2.99·18-s − 7.29·19-s − 4.12·20-s + 0.00657·21-s − 1.51·22-s + 6.22·23-s + 0.00489·24-s + 12.0·25-s − 3.47·26-s + 0.0293·27-s − 1.34·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.00282·3-s + 0.5·4-s − 1.84·5-s + 0.00199·6-s − 0.507·7-s − 0.353·8-s − 0.999·9-s + 1.30·10-s + 0.457·11-s − 0.00141·12-s + 0.963·13-s + 0.359·14-s + 0.00521·15-s + 0.250·16-s + 0.220·17-s + 0.707·18-s − 1.67·19-s − 0.922·20-s + 0.00143·21-s − 0.323·22-s + 1.29·23-s + 0.000998·24-s + 2.40·25-s − 0.681·26-s + 0.00564·27-s − 0.253·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 0.00489T + 3T^{2} \) |
| 5 | \( 1 + 4.12T + 5T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 0.907T + 17T^{2} \) |
| 19 | \( 1 + 7.29T + 19T^{2} \) |
| 23 | \( 1 - 6.22T + 23T^{2} \) |
| 29 | \( 1 - 4.77T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 + 2.90T + 37T^{2} \) |
| 41 | \( 1 + 0.820T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 - 5.64T + 47T^{2} \) |
| 53 | \( 1 + 5.23T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.89T + 61T^{2} \) |
| 67 | \( 1 + 4.82T + 67T^{2} \) |
| 71 | \( 1 + 3.85T + 71T^{2} \) |
| 73 | \( 1 + 2.23T + 73T^{2} \) |
| 79 | \( 1 - 2.39T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 1.14T + 89T^{2} \) |
| 97 | \( 1 - 7.39T + 97T^{2} \) |
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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
−8.210539980669527758965603432361, −7.51610272362539352002287956700, −6.68218585390921914387965994786, −6.21094610140753290954024955659, −5.00860412797978572536439468004, −4.00296383026875964879793061695, −3.43650220728290469018705617343, −2.60285604401887591863943729457, −0.989370740402425403024770637418, 0,
0.989370740402425403024770637418, 2.60285604401887591863943729457, 3.43650220728290469018705617343, 4.00296383026875964879793061695, 5.00860412797978572536439468004, 6.21094610140753290954024955659, 6.68218585390921914387965994786, 7.51610272362539352002287956700, 8.210539980669527758965603432361
