| L(s) = 1 | + 2-s − 3-s + 4-s + 3.05·5-s − 6-s + 7-s + 8-s + 9-s + 3.05·10-s + 3.44·11-s − 12-s + 14-s − 3.05·15-s + 16-s − 2.82·17-s + 18-s − 1.72·19-s + 3.05·20-s − 21-s + 3.44·22-s + 3.06·23-s − 24-s + 4.33·25-s − 27-s + 28-s + 3.85·29-s − 3.05·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.36·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.966·10-s + 1.03·11-s − 0.288·12-s + 0.267·14-s − 0.789·15-s + 0.250·16-s − 0.686·17-s + 0.235·18-s − 0.395·19-s + 0.683·20-s − 0.218·21-s + 0.733·22-s + 0.638·23-s − 0.204·24-s + 0.867·25-s − 0.192·27-s + 0.188·28-s + 0.715·29-s − 0.557·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.186632894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.186632894\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.05T + 5T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 - 0.978T + 31T^{2} \) |
| 37 | \( 1 + 5.29T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 - 8.04T + 47T^{2} \) |
| 53 | \( 1 + 8.33T + 53T^{2} \) |
| 59 | \( 1 - 9.97T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4.75T + 67T^{2} \) |
| 71 | \( 1 - 3.66T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 4.49T + 79T^{2} \) |
| 83 | \( 1 + 7.15T + 83T^{2} \) |
| 89 | \( 1 - 7.90T + 89T^{2} \) |
| 97 | \( 1 - 9.09T + 97T^{2} \) |
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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
−7.74470915063006466268317032391, −6.83662042004929040220914741966, −6.41209518334899231722715532455, −5.87241052665359920388725587744, −5.11457964814019012630648343705, −4.54616217731799260290715779015, −3.73111852219833090228760511140, −2.59649929374268854069918597742, −1.87157100759576888755209541111, −1.02898520787045387045439282929,
1.02898520787045387045439282929, 1.87157100759576888755209541111, 2.59649929374268854069918597742, 3.73111852219833090228760511140, 4.54616217731799260290715779015, 5.11457964814019012630648343705, 5.87241052665359920388725587744, 6.41209518334899231722715532455, 6.83662042004929040220914741966, 7.74470915063006466268317032391
