| L(s) = 1 | + 2.56·2-s + 4.56·4-s − 5-s − 0.860·7-s + 6.58·8-s − 2.56·10-s + 4.21·11-s + 1.77·13-s − 2.20·14-s + 7.73·16-s + 3.70·17-s + 0.190·19-s − 4.56·20-s + 10.7·22-s − 5.54·23-s + 25-s + 4.55·26-s − 3.93·28-s + 5.89·29-s − 1.66·31-s + 6.66·32-s + 9.49·34-s + 0.860·35-s − 2.99·37-s + 0.489·38-s − 6.58·40-s + 5.52·41-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 2.28·4-s − 0.447·5-s − 0.325·7-s + 2.32·8-s − 0.810·10-s + 1.26·11-s + 0.493·13-s − 0.589·14-s + 1.93·16-s + 0.898·17-s + 0.0437·19-s − 1.02·20-s + 2.30·22-s − 1.15·23-s + 0.200·25-s + 0.893·26-s − 0.742·28-s + 1.09·29-s − 0.298·31-s + 1.17·32-s + 1.62·34-s + 0.145·35-s − 0.491·37-s + 0.0793·38-s − 1.04·40-s + 0.862·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.248604798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.248604798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 7 | \( 1 + 0.860T + 7T^{2} \) |
| 11 | \( 1 - 4.21T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 0.190T + 19T^{2} \) |
| 23 | \( 1 + 5.54T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 + 2.99T + 37T^{2} \) |
| 41 | \( 1 - 5.52T + 41T^{2} \) |
| 43 | \( 1 - 9.56T + 43T^{2} \) |
| 47 | \( 1 - 0.668T + 47T^{2} \) |
| 53 | \( 1 + 8.18T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 7.19T + 61T^{2} \) |
| 67 | \( 1 + 6.33T + 67T^{2} \) |
| 71 | \( 1 + 4.00T + 71T^{2} \) |
| 73 | \( 1 - 5.79T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 6.87T + 83T^{2} \) |
| 97 | \( 1 - 8.89T + 97T^{2} \) |
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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
−8.195927028525966173702361314988, −7.43651607625128883317535745506, −6.61672641066152397493796905713, −6.16285544763584486039797819233, −5.43919014202523038058585082480, −4.50248908795820181815506856738, −3.85048234159538605588068033601, −3.39085582259242869524867735171, −2.36690659292948438868951785923, −1.19512460880325834628769243433,
1.19512460880325834628769243433, 2.36690659292948438868951785923, 3.39085582259242869524867735171, 3.85048234159538605588068033601, 4.50248908795820181815506856738, 5.43919014202523038058585082480, 6.16285544763584486039797819233, 6.61672641066152397493796905713, 7.43651607625128883317535745506, 8.195927028525966173702361314988
