| L(s) = 1 | − 3.82·2-s + 6.65·4-s + 7·7-s + 5.14·8-s − 48.5·11-s + 43.6·13-s − 26.7·14-s − 72.9·16-s − 67.6·17-s − 93.2·19-s + 185.·22-s − 104.·23-s − 167.·26-s + 46.5·28-s + 58.7·29-s − 9.08·31-s + 238.·32-s + 259.·34-s + 252.·37-s + 357.·38-s − 276.·41-s + 92.6·43-s − 323.·44-s + 398.·46-s − 582.·47-s + 49·49-s + 290.·52-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.832·4-s + 0.377·7-s + 0.227·8-s − 1.33·11-s + 0.931·13-s − 0.511·14-s − 1.13·16-s − 0.965·17-s − 1.12·19-s + 1.80·22-s − 0.944·23-s − 1.26·26-s + 0.314·28-s + 0.376·29-s − 0.0526·31-s + 1.31·32-s + 1.30·34-s + 1.12·37-s + 1.52·38-s − 1.05·41-s + 0.328·43-s − 1.10·44-s + 1.27·46-s − 1.80·47-s + 0.142·49-s + 0.775·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5786894662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5786894662\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 3.82T + 8T^{2} \) |
| 11 | \( 1 + 48.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 67.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 58.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.08T + 2.97e4T^{2} \) |
| 37 | \( 1 - 252.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 92.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 582.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 623.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 524.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 352.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 736.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 492.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 872.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 529.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 463.T + 9.12e5T^{2} \) |
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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
−8.794681458740001325033186038992, −8.413142498911376468579965383861, −7.81737804189941971700052575388, −6.87848038655235803818136254248, −6.03663086875579142519159751243, −4.89630613255668499624331639274, −4.04163809891441802831668625980, −2.53948128548143492600136155468, −1.73025895600324539075960712269, −0.44771854296029226229800340617,
0.44771854296029226229800340617, 1.73025895600324539075960712269, 2.53948128548143492600136155468, 4.04163809891441802831668625980, 4.89630613255668499624331639274, 6.03663086875579142519159751243, 6.87848038655235803818136254248, 7.81737804189941971700052575388, 8.413142498911376468579965383861, 8.794681458740001325033186038992
