| L(s) = 1 | + 2.90·3-s + 3.52·7-s + 5.42·9-s − 3.80·11-s + 2.62·13-s − 5.80·17-s + 5.05·19-s + 10.2·21-s + 0.474·23-s + 7.05·27-s + 2·29-s + 2.75·31-s − 11.0·33-s + 7.18·37-s + 7.61·39-s + 5.18·41-s − 1.95·43-s − 5.33·47-s + 5.42·49-s − 16.8·51-s + 5.37·53-s + 14.6·57-s − 5.05·59-s + 12.2·61-s + 19.1·63-s − 7.76·67-s + 1.37·69-s + ⋯ |
| L(s) = 1 | + 1.67·3-s + 1.33·7-s + 1.80·9-s − 1.14·11-s + 0.727·13-s − 1.40·17-s + 1.15·19-s + 2.23·21-s + 0.0989·23-s + 1.35·27-s + 0.371·29-s + 0.494·31-s − 1.92·33-s + 1.18·37-s + 1.21·39-s + 0.809·41-s − 0.297·43-s − 0.777·47-s + 0.775·49-s − 2.36·51-s + 0.738·53-s + 1.94·57-s − 0.657·59-s + 1.56·61-s + 2.41·63-s − 0.948·67-s + 0.165·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.066834293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.066834293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 - 0.474T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 + 1.95T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 + 5.05T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 7.76T + 67T^{2} \) |
| 71 | \( 1 - 4.85T + 71T^{2} \) |
| 73 | \( 1 - 6.66T + 73T^{2} \) |
| 79 | \( 1 - 5.24T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.409994946851079660743051549445, −8.127765198850310466331657523727, −7.53989125515452335850629359849, −6.66350414592702488440193552197, −5.42587236644732432309727528510, −4.64016085892759175923763874544, −3.91656470800414672038494014557, −2.84078851710971833436420549251, −2.27337280958736637457542518693, −1.24857163045626404977341858825,
1.24857163045626404977341858825, 2.27337280958736637457542518693, 2.84078851710971833436420549251, 3.91656470800414672038494014557, 4.64016085892759175923763874544, 5.42587236644732432309727528510, 6.66350414592702488440193552197, 7.53989125515452335850629359849, 8.127765198850310466331657523727, 8.409994946851079660743051549445
