| L(s) = 1 | − 1.36·3-s − 5-s + 2.86·7-s − 1.12·9-s + 1.14·11-s + 0.207·13-s + 1.36·15-s + 5.28·17-s + 8.39·19-s − 3.91·21-s + 7.69·23-s + 25-s + 5.64·27-s + 0.349·29-s + 7.66·31-s − 1.57·33-s − 2.86·35-s − 9.65·37-s − 0.283·39-s − 3.06·41-s + 8.05·43-s + 1.12·45-s + 5.25·47-s + 1.19·49-s − 7.22·51-s − 9.65·53-s − 1.14·55-s + ⋯ |
| L(s) = 1 | − 0.789·3-s − 0.447·5-s + 1.08·7-s − 0.376·9-s + 0.346·11-s + 0.0575·13-s + 0.353·15-s + 1.28·17-s + 1.92·19-s − 0.854·21-s + 1.60·23-s + 0.200·25-s + 1.08·27-s + 0.0648·29-s + 1.37·31-s − 0.273·33-s − 0.483·35-s − 1.58·37-s − 0.0454·39-s − 0.479·41-s + 1.22·43-s + 0.168·45-s + 0.766·47-s + 0.170·49-s − 1.01·51-s − 1.32·53-s − 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.917715063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917715063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 1.36T + 3T^{2} \) |
| 7 | \( 1 - 2.86T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 - 0.207T + 13T^{2} \) |
| 17 | \( 1 - 5.28T + 17T^{2} \) |
| 19 | \( 1 - 8.39T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 - 0.349T + 29T^{2} \) |
| 31 | \( 1 - 7.66T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 + 3.06T + 41T^{2} \) |
| 43 | \( 1 - 8.05T + 43T^{2} \) |
| 47 | \( 1 - 5.25T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 - 0.336T + 59T^{2} \) |
| 61 | \( 1 + 1.01T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 + 1.96T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 4.88T + 83T^{2} \) |
| 89 | \( 1 + 0.560T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.74404916022513263049196148896, −7.22961469078501417536667460859, −6.43575530126029018703538969519, −5.49309809182460200928744251425, −5.18251114585581826398899729174, −4.54123913062409507454978188370, −3.41699607286336194473621988093, −2.87223120031518094433831352868, −1.38094843564984923079460361079, −0.826439347284070686466780932166,
0.826439347284070686466780932166, 1.38094843564984923079460361079, 2.87223120031518094433831352868, 3.41699607286336194473621988093, 4.54123913062409507454978188370, 5.18251114585581826398899729174, 5.49309809182460200928744251425, 6.43575530126029018703538969519, 7.22961469078501417536667460859, 7.74404916022513263049196148896
