| L(s) = 1 | + 1.01·3-s − 1.13·5-s − 4.18·7-s − 1.96·9-s − 2.25·11-s − 5.91·13-s − 1.15·15-s − 1.01·17-s + 2.69·19-s − 4.25·21-s + 3.02·23-s − 3.70·25-s − 5.04·27-s + 1.59·29-s − 9.38·31-s − 2.29·33-s + 4.75·35-s − 9.18·37-s − 6.01·39-s + 10.8·41-s + 4.68·43-s + 2.23·45-s + 5.29·47-s + 10.4·49-s − 1.02·51-s − 2.86·53-s + 2.56·55-s + ⋯ |
| L(s) = 1 | + 0.586·3-s − 0.508·5-s − 1.58·7-s − 0.655·9-s − 0.680·11-s − 1.64·13-s − 0.298·15-s − 0.245·17-s + 0.617·19-s − 0.927·21-s + 0.631·23-s − 0.741·25-s − 0.971·27-s + 0.295·29-s − 1.68·31-s − 0.399·33-s + 0.803·35-s − 1.50·37-s − 0.962·39-s + 1.69·41-s + 0.715·43-s + 0.333·45-s + 0.772·47-s + 1.49·49-s − 0.143·51-s − 0.392·53-s + 0.345·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5347910557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5347910557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 383 | \( 1 + T \) |
| good | 3 | \( 1 - 1.01T + 3T^{2} \) |
| 5 | \( 1 + 1.13T + 5T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 + 2.25T + 11T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 + 1.01T + 17T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 23 | \( 1 - 3.02T + 23T^{2} \) |
| 29 | \( 1 - 1.59T + 29T^{2} \) |
| 31 | \( 1 + 9.38T + 31T^{2} \) |
| 37 | \( 1 + 9.18T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 + 2.86T + 53T^{2} \) |
| 59 | \( 1 - 9.98T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 2.17T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 8.14T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 2.61T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 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
−7.947337902531973112257096684144, −7.32940164434354235334628951659, −6.94219991023897863671679770907, −5.78881825119052746104751935932, −5.36269743560679464568546048326, −4.25857108771159457333610297385, −3.42977949156104238550900826379, −2.85403607057022271233078147350, −2.21767935603171068150518302175, −0.33859756096322468965390353139,
0.33859756096322468965390353139, 2.21767935603171068150518302175, 2.85403607057022271233078147350, 3.42977949156104238550900826379, 4.25857108771159457333610297385, 5.36269743560679464568546048326, 5.78881825119052746104751935932, 6.94219991023897863671679770907, 7.32940164434354235334628951659, 7.947337902531973112257096684144
