| L(s) = 1 | − 0.347·2-s + 3-s − 1.87·4-s − 3.53·5-s − 0.347·6-s + 7-s + 1.34·8-s + 9-s + 1.22·10-s + 3.29·11-s − 1.87·12-s − 0.347·14-s − 3.53·15-s + 3.29·16-s − 5.17·17-s − 0.347·18-s − 6.06·19-s + 6.63·20-s + 21-s − 1.14·22-s + 1.49·23-s + 1.34·24-s + 7.47·25-s + 27-s − 1.87·28-s + 4.73·29-s + 1.22·30-s + ⋯ |
| L(s) = 1 | − 0.245·2-s + 0.577·3-s − 0.939·4-s − 1.57·5-s − 0.141·6-s + 0.377·7-s + 0.476·8-s + 0.333·9-s + 0.387·10-s + 0.992·11-s − 0.542·12-s − 0.0928·14-s − 0.911·15-s + 0.822·16-s − 1.25·17-s − 0.0818·18-s − 1.39·19-s + 1.48·20-s + 0.218·21-s − 0.243·22-s + 0.310·23-s + 0.275·24-s + 1.49·25-s + 0.192·27-s − 0.355·28-s + 0.879·29-s + 0.223·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.347T + 2T^{2} \) |
| 5 | \( 1 + 3.53T + 5T^{2} \) |
| 11 | \( 1 - 3.29T + 11T^{2} \) |
| 17 | \( 1 + 5.17T + 17T^{2} \) |
| 19 | \( 1 + 6.06T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 - 1.59T + 31T^{2} \) |
| 37 | \( 1 + 7.90T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 0.162T + 47T^{2} \) |
| 53 | \( 1 - 6.84T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 1.82T + 61T^{2} \) |
| 67 | \( 1 + 9.18T + 67T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 9.22T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 0.0641T + 89T^{2} \) |
| 97 | \( 1 + 0.448T + 97T^{2} \) |
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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
−8.425790907263548414757188658436, −7.59457808420896218305924969899, −7.03320543815686493189970065785, −6.03631889455478772097577523709, −4.59654493855891541992817994342, −4.32828257515971463709890730770, −3.78086655075197991765647044541, −2.62345699853085200528484152390, −1.22093260495573513554144322579, 0,
1.22093260495573513554144322579, 2.62345699853085200528484152390, 3.78086655075197991765647044541, 4.32828257515971463709890730770, 4.59654493855891541992817994342, 6.03631889455478772097577523709, 7.03320543815686493189970065785, 7.59457808420896218305924969899, 8.425790907263548414757188658436
