L(s) = 1 | + 0.685·2-s − 7.53·4-s + 9.28·5-s − 6.04·7-s − 10.6·8-s + 6.35·10-s + 59.2·11-s − 4.14·14-s + 52.9·16-s − 117.·17-s − 49.2·19-s − 69.9·20-s + 40.5·22-s + 23.1·23-s − 38.8·25-s + 45.5·28-s − 145.·29-s + 94.9·31-s + 121.·32-s − 80.4·34-s − 56.1·35-s + 379.·37-s − 33.7·38-s − 98.7·40-s − 268.·41-s − 23.6·43-s − 445.·44-s + ⋯ |
L(s) = 1 | + 0.242·2-s − 0.941·4-s + 0.830·5-s − 0.326·7-s − 0.470·8-s + 0.201·10-s + 1.62·11-s − 0.0790·14-s + 0.827·16-s − 1.67·17-s − 0.594·19-s − 0.781·20-s + 0.393·22-s + 0.209·23-s − 0.310·25-s + 0.307·28-s − 0.928·29-s + 0.550·31-s + 0.670·32-s − 0.405·34-s − 0.271·35-s + 1.68·37-s − 0.144·38-s − 0.390·40-s − 1.02·41-s − 0.0839·43-s − 1.52·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.685T + 8T^{2} \) |
| 5 | \( 1 - 9.28T + 125T^{2} \) |
| 7 | \( 1 + 6.04T + 343T^{2} \) |
| 11 | \( 1 - 59.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 23.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 94.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 379.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 268.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 23.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 301.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 391.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 510.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 520.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 470.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 466.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 314.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 47.9T + 4.93e5T^{2} \) |
| 83 | \( 1 - 310.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 216.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 219.T + 9.12e5T^{2} \) |
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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
−8.971070532177356961968434988442, −8.117433220884151689941201708507, −6.70948557054096937126846013699, −6.32382290479481289864714673548, −5.38412116904312951512151880666, −4.33980385194299857643188663753, −3.82628300972319363151770966258, −2.47399142394953992691051514873, −1.33334161540118623206424438797, 0,
1.33334161540118623206424438797, 2.47399142394953992691051514873, 3.82628300972319363151770966258, 4.33980385194299857643188663753, 5.38412116904312951512151880666, 6.32382290479481289864714673548, 6.70948557054096937126846013699, 8.117433220884151689941201708507, 8.971070532177356961968434988442