| L(s) = 1 | − 2-s + 0.838·3-s + 4-s − 3.46·5-s − 0.838·6-s + 1.66·7-s − 8-s − 2.29·9-s + 3.46·10-s − 4.85·11-s + 0.838·12-s − 1.45·13-s − 1.66·14-s − 2.90·15-s + 16-s − 6.56·17-s + 2.29·18-s + 19-s − 3.46·20-s + 1.39·21-s + 4.85·22-s − 4.88·23-s − 0.838·24-s + 7.03·25-s + 1.45·26-s − 4.44·27-s + 1.66·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.484·3-s + 0.5·4-s − 1.55·5-s − 0.342·6-s + 0.628·7-s − 0.353·8-s − 0.765·9-s + 1.09·10-s − 1.46·11-s + 0.242·12-s − 0.404·13-s − 0.444·14-s − 0.751·15-s + 0.250·16-s − 1.59·17-s + 0.541·18-s + 0.229·19-s − 0.775·20-s + 0.304·21-s + 1.03·22-s − 1.01·23-s − 0.171·24-s + 1.40·25-s + 0.286·26-s − 0.854·27-s + 0.314·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08392322263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08392322263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 0.838T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 1.66T + 7T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 + 6.56T + 17T^{2} \) |
| 23 | \( 1 + 4.88T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 3.08T + 47T^{2} \) |
| 53 | \( 1 + 0.453T + 53T^{2} \) |
| 59 | \( 1 + 1.59T + 59T^{2} \) |
| 61 | \( 1 - 0.814T + 61T^{2} \) |
| 67 | \( 1 + 9.72T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 1.57T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−7.87543973347683299475587066187, −7.51227769347975641041465493557, −6.80500606966411741243706569503, −5.72754981538134179442069236724, −4.96654968346063458789374118323, −4.22100375502988476523305260684, −3.37046273600195216453177313460, −2.60980298623241825162031405402, −1.87398514670607995058832922076, −0.14420409364077784041042937092,
0.14420409364077784041042937092, 1.87398514670607995058832922076, 2.60980298623241825162031405402, 3.37046273600195216453177313460, 4.22100375502988476523305260684, 4.96654968346063458789374118323, 5.72754981538134179442069236724, 6.80500606966411741243706569503, 7.51227769347975641041465493557, 7.87543973347683299475587066187
