| L(s) = 1 | − 5.42·2-s − 3·3-s + 21.4·4-s + 16.2·6-s + 10.2·7-s − 73.0·8-s + 9·9-s + 12.6·11-s − 64.3·12-s − 84.2·13-s − 55.5·14-s + 224.·16-s − 6.89·17-s − 48.8·18-s − 79.7·19-s − 30.7·21-s − 68.5·22-s + 73.8·23-s + 219.·24-s + 457.·26-s − 27·27-s + 219.·28-s + 29·29-s + 0.254·31-s − 635.·32-s − 37.8·33-s + 37.4·34-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 0.577·3-s + 2.68·4-s + 1.10·6-s + 0.552·7-s − 3.22·8-s + 0.333·9-s + 0.346·11-s − 1.54·12-s − 1.79·13-s − 1.06·14-s + 3.51·16-s − 0.0983·17-s − 0.639·18-s − 0.962·19-s − 0.319·21-s − 0.664·22-s + 0.669·23-s + 1.86·24-s + 3.44·26-s − 0.192·27-s + 1.48·28-s + 0.185·29-s + 0.00147·31-s − 3.50·32-s − 0.199·33-s + 0.188·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 5.42T + 8T^{2} \) |
| 7 | \( 1 - 10.2T + 343T^{2} \) |
| 11 | \( 1 - 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 84.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.89T + 4.91e3T^{2} \) |
| 19 | \( 1 + 79.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 73.8T + 1.21e4T^{2} \) |
| 31 | \( 1 - 0.254T + 2.97e4T^{2} \) |
| 37 | \( 1 + 40.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 208.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 194.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 522.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 294.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 196.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 189.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 570.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 255.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 983.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 678.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 30.8T + 9.12e5T^{2} \) |
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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
−8.418937203739612021135927028249, −7.55253070829367460581127045146, −7.12633880512948888075913936775, −6.33328659835659915858589921485, −5.39467360992848045134459411231, −4.33457025699273684932961425694, −2.72744975553151661830918056752, −1.99582895478742579706885361553, −0.926431980759268176656840263903, 0,
0.926431980759268176656840263903, 1.99582895478742579706885361553, 2.72744975553151661830918056752, 4.33457025699273684932961425694, 5.39467360992848045134459411231, 6.33328659835659915858589921485, 7.12633880512948888075913936775, 7.55253070829367460581127045146, 8.418937203739612021135927028249
