| L(s) = 1 | + 2-s − 1.56·3-s + 4-s − 1.56·6-s + 2.56·7-s + 8-s − 0.561·9-s − 11-s − 1.56·12-s − 0.561·13-s + 2.56·14-s + 16-s + 5.56·17-s − 0.561·18-s + 3·19-s − 4·21-s − 22-s + 23-s − 1.56·24-s − 0.561·26-s + 5.56·27-s + 2.56·28-s + 1.43·29-s − 5.12·31-s + 32-s + 1.56·33-s + 5.56·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.901·3-s + 0.5·4-s − 0.637·6-s + 0.968·7-s + 0.353·8-s − 0.187·9-s − 0.301·11-s − 0.450·12-s − 0.155·13-s + 0.684·14-s + 0.250·16-s + 1.34·17-s − 0.132·18-s + 0.688·19-s − 0.872·21-s − 0.213·22-s + 0.208·23-s − 0.318·24-s − 0.110·26-s + 1.07·27-s + 0.484·28-s + 0.267·29-s − 0.920·31-s + 0.176·32-s + 0.271·33-s + 0.953·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.059158258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.059158258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 0.561T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 29 | \( 1 - 1.43T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 9.12T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2.24T + 61T^{2} \) |
| 67 | \( 1 + 5.56T + 67T^{2} \) |
| 71 | \( 1 - 1.12T + 71T^{2} \) |
| 73 | \( 1 + 6.12T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 - 6.12T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−10.12692308139572891503609325885, −8.956393688436070180469627443793, −7.88499122980998668561252838126, −7.29550164266607292843466150076, −6.12569378464105356367199929440, −5.39775972762451837821076117968, −4.94880345770343995077458555090, −3.75908119626938310371028282366, −2.56470085420555523446582119857, −1.08862002845527615608999315659,
1.08862002845527615608999315659, 2.56470085420555523446582119857, 3.75908119626938310371028282366, 4.94880345770343995077458555090, 5.39775972762451837821076117968, 6.12569378464105356367199929440, 7.29550164266607292843466150076, 7.88499122980998668561252838126, 8.956393688436070180469627443793, 10.12692308139572891503609325885
