| L(s) = 1 | + 2.44·2-s − 3-s + 3.95·4-s − 2.44·6-s − 3.44·7-s + 4.77·8-s + 9-s − 3.26·11-s − 3.95·12-s − 3.23·13-s − 8.39·14-s + 3.73·16-s − 5.05·17-s + 2.44·18-s − 3.08·19-s + 3.44·21-s − 7.97·22-s + 1.54·23-s − 4.77·24-s − 7.88·26-s − 27-s − 13.6·28-s − 3.12·29-s + 7.44·31-s − 0.434·32-s + 3.26·33-s − 12.3·34-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.577·3-s + 1.97·4-s − 0.996·6-s − 1.30·7-s + 1.68·8-s + 0.333·9-s − 0.984·11-s − 1.14·12-s − 0.896·13-s − 2.24·14-s + 0.932·16-s − 1.22·17-s + 0.575·18-s − 0.706·19-s + 0.750·21-s − 1.69·22-s + 0.322·23-s − 0.973·24-s − 1.54·26-s − 0.192·27-s − 2.57·28-s − 0.579·29-s + 1.33·31-s − 0.0768·32-s + 0.568·33-s − 2.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 - 1.54T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 - 5.41T + 41T^{2} \) |
| 43 | \( 1 - 2.53T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 1.73T + 59T^{2} \) |
| 61 | \( 1 - 7.83T + 61T^{2} \) |
| 67 | \( 1 + 1.84T + 67T^{2} \) |
| 71 | \( 1 + 0.713T + 71T^{2} \) |
| 73 | \( 1 + 1.88T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 3.95T + 83T^{2} \) |
| 89 | \( 1 - 8.53T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.910754217724036212071170006981, −7.58508248957203711199407095975, −6.80574714303647897187476783830, −6.30499807210943481816425291427, −5.50994534556725166603312522879, −4.74356616171398873793149577855, −4.03638071102947295065332022120, −2.93468732331815436161634602199, −2.29417566796639957595857191229, 0,
2.29417566796639957595857191229, 2.93468732331815436161634602199, 4.03638071102947295065332022120, 4.74356616171398873793149577855, 5.50994534556725166603312522879, 6.30499807210943481816425291427, 6.80574714303647897187476783830, 7.58508248957203711199407095975, 8.910754217724036212071170006981
