| L(s) = 1 | + 2-s − 2.35·3-s + 4-s − 2.35·6-s − 4.49·7-s + 8-s + 2.55·9-s − 2.69·11-s − 2.35·12-s − 4.49·14-s + 16-s − 3.58·17-s + 2.55·18-s + 2.93·19-s + 10.5·21-s − 2.69·22-s + 6.09·23-s − 2.35·24-s + 1.04·27-s − 4.49·28-s + 2.98·29-s + 2.39·31-s + 32-s + 6.34·33-s − 3.58·34-s + 2.55·36-s − 1.50·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.36·3-s + 0.5·4-s − 0.962·6-s − 1.69·7-s + 0.353·8-s + 0.851·9-s − 0.811·11-s − 0.680·12-s − 1.20·14-s + 0.250·16-s − 0.868·17-s + 0.602·18-s + 0.674·19-s + 2.31·21-s − 0.573·22-s + 1.27·23-s − 0.481·24-s + 0.201·27-s − 0.849·28-s + 0.554·29-s + 0.430·31-s + 0.176·32-s + 1.10·33-s − 0.614·34-s + 0.425·36-s − 0.247·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 7 | \( 1 + 4.49T + 7T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 17 | \( 1 + 3.58T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 - 2.98T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 37 | \( 1 + 1.50T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 + 0.170T + 43T^{2} \) |
| 47 | \( 1 - 5.20T + 47T^{2} \) |
| 53 | \( 1 + 6.09T + 53T^{2} \) |
| 59 | \( 1 + 3.07T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 2.81T + 79T^{2} \) |
| 83 | \( 1 - 2.93T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.03855913331863526800653200061, −6.51277994630628964276953137305, −6.15053939034626243765735233857, −5.29846128669169331576167364465, −4.91846899235010224905144715349, −3.97743452905997139928619336139, −3.08513742841735784961134948839, −2.53431299275718223173148661093, −0.973642549701648523028651052190, 0,
0.973642549701648523028651052190, 2.53431299275718223173148661093, 3.08513742841735784961134948839, 3.97743452905997139928619336139, 4.91846899235010224905144715349, 5.29846128669169331576167364465, 6.15053939034626243765735233857, 6.51277994630628964276953137305, 7.03855913331863526800653200061
