| L(s) = 1 | + 4.95·2-s − 3·3-s + 16.5·4-s − 14.8·6-s − 28.2·7-s + 42.1·8-s + 9·9-s − 9.92·11-s − 49.5·12-s + 92.2·13-s − 140.·14-s + 76.7·16-s + 19.6·17-s + 44.5·18-s − 76.9·19-s + 84.8·21-s − 49.1·22-s − 101.·23-s − 126.·24-s + 456.·26-s − 27·27-s − 467.·28-s + 216.·29-s + 105.·31-s + 42.6·32-s + 29.7·33-s + 97.1·34-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.577·3-s + 2.06·4-s − 1.01·6-s − 1.52·7-s + 1.86·8-s + 0.333·9-s − 0.271·11-s − 1.19·12-s + 1.96·13-s − 2.67·14-s + 1.19·16-s + 0.279·17-s + 0.583·18-s − 0.928·19-s + 0.881·21-s − 0.476·22-s − 0.918·23-s − 1.07·24-s + 3.44·26-s − 0.192·27-s − 3.15·28-s + 1.38·29-s + 0.608·31-s + 0.235·32-s + 0.156·33-s + 0.489·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.853375520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.853375520\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 4.95T + 8T^{2} \) |
| 7 | \( 1 + 28.2T + 343T^{2} \) |
| 11 | \( 1 + 9.92T + 1.33e3T^{2} \) |
| 13 | \( 1 - 92.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 101.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 105.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 331.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 95.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 67.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 117.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 222.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 174.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 472.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 125.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 927.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 101.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 295.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 127.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.52e3T + 9.12e5T^{2} \) |
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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
−8.849025005556927520629094880846, −7.81356390957485816449543676163, −6.57451895663044286516164868235, −6.26770250581689766769952519305, −5.87285357157843783303565048500, −4.70754284467643380745781864786, −3.90343230841541464321591916510, −3.30676733063092464971939496568, −2.31089524491549898163031506580, −0.802502945670443288750964174482,
0.802502945670443288750964174482, 2.31089524491549898163031506580, 3.30676733063092464971939496568, 3.90343230841541464321591916510, 4.70754284467643380745781864786, 5.87285357157843783303565048500, 6.26770250581689766769952519305, 6.57451895663044286516164868235, 7.81356390957485816449543676163, 8.849025005556927520629094880846
