| L(s) = 1 | + 2-s + 3.30·3-s + 4-s + 0.0221·5-s + 3.30·6-s + 1.86·7-s + 8-s + 7.93·9-s + 0.0221·10-s + 11-s + 3.30·12-s + 6.94·13-s + 1.86·14-s + 0.0731·15-s + 16-s − 7.96·17-s + 7.93·18-s + 6.05·19-s + 0.0221·20-s + 6.15·21-s + 22-s − 7.18·23-s + 3.30·24-s − 4.99·25-s + 6.94·26-s + 16.3·27-s + 1.86·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.90·3-s + 0.5·4-s + 0.00989·5-s + 1.34·6-s + 0.703·7-s + 0.353·8-s + 2.64·9-s + 0.00699·10-s + 0.301·11-s + 0.954·12-s + 1.92·13-s + 0.497·14-s + 0.0188·15-s + 0.250·16-s − 1.93·17-s + 1.86·18-s + 1.38·19-s + 0.00494·20-s + 1.34·21-s + 0.213·22-s − 1.49·23-s + 0.674·24-s − 0.999·25-s + 1.36·26-s + 3.13·27-s + 0.351·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.310448101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.310448101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 - 0.0221T + 5T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 13 | \( 1 - 6.94T + 13T^{2} \) |
| 17 | \( 1 + 7.96T + 17T^{2} \) |
| 19 | \( 1 - 6.05T + 19T^{2} \) |
| 23 | \( 1 + 7.18T + 23T^{2} \) |
| 29 | \( 1 - 2.19T + 29T^{2} \) |
| 31 | \( 1 - 0.533T + 31T^{2} \) |
| 37 | \( 1 + 8.74T + 37T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 0.484T + 53T^{2} \) |
| 59 | \( 1 + 1.90T + 59T^{2} \) |
| 61 | \( 1 - 3.89T + 61T^{2} \) |
| 67 | \( 1 - 7.36T + 67T^{2} \) |
| 71 | \( 1 - 1.84T + 71T^{2} \) |
| 73 | \( 1 - 8.56T + 73T^{2} \) |
| 79 | \( 1 + 9.80T + 79T^{2} \) |
| 83 | \( 1 + 1.01T + 83T^{2} \) |
| 89 | \( 1 - 5.18T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.333010611010690625672856221882, −7.939261659655220614989164989318, −6.84888123750274443305415933247, −6.43478678835518666186964721425, −5.17017328265493649415649659583, −4.32183167605712553805834923625, −3.66361933284509721784523692206, −3.19531338634851418047607979812, −1.86386962416674926994638750922, −1.67646706980565778738227820202,
1.67646706980565778738227820202, 1.86386962416674926994638750922, 3.19531338634851418047607979812, 3.66361933284509721784523692206, 4.32183167605712553805834923625, 5.17017328265493649415649659583, 6.43478678835518666186964721425, 6.84888123750274443305415933247, 7.939261659655220614989164989318, 8.333010611010690625672856221882
