| L(s) = 1 | − 1.33·2-s − 3·3-s − 6.20·4-s + 4.01·6-s + 12.2·7-s + 19.0·8-s + 9·9-s − 2.48·11-s + 18.6·12-s − 85.6·13-s − 16.3·14-s + 24.1·16-s + 5.54·17-s − 12.0·18-s − 24.1·19-s − 36.6·21-s + 3.32·22-s + 103.·23-s − 57.0·24-s + 114.·26-s − 27·27-s − 75.7·28-s − 114.·29-s + 153.·31-s − 184.·32-s + 7.44·33-s − 7.43·34-s + ⋯ |
| L(s) = 1 | − 0.473·2-s − 0.577·3-s − 0.775·4-s + 0.273·6-s + 0.659·7-s + 0.841·8-s + 0.333·9-s − 0.0680·11-s + 0.447·12-s − 1.82·13-s − 0.312·14-s + 0.377·16-s + 0.0791·17-s − 0.157·18-s − 0.291·19-s − 0.380·21-s + 0.0322·22-s + 0.936·23-s − 0.485·24-s + 0.865·26-s − 0.192·27-s − 0.511·28-s − 0.731·29-s + 0.891·31-s − 1.01·32-s + 0.0392·33-s − 0.0374·34-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
| \(L(2)\) |
\(\approx\) |
\(0.7393570705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7393570705\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.33T + 8T^{2} \) |
| 7 | \( 1 - 12.2T + 343T^{2} \) |
| 11 | \( 1 + 2.48T + 1.33e3T^{2} \) |
| 13 | \( 1 + 85.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 24.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 114.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 153.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 342.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 34.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 645.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 662.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 357.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 531.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 634.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 962.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 568.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 319.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 399.T + 9.12e5T^{2} \) |
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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
−8.987177181849458777860760329529, −7.954039143402871806607944942063, −7.56447096896047947573525794487, −6.58593692544131047267409907639, −5.42141789418775952189310864756, −4.82073725464271040496124234242, −4.25408101646129624305365956803, −2.79273654369833676871940003519, −1.56468651965471053649777396462, −0.46484314305370674915485829899,
0.46484314305370674915485829899, 1.56468651965471053649777396462, 2.79273654369833676871940003519, 4.25408101646129624305365956803, 4.82073725464271040496124234242, 5.42141789418775952189310864756, 6.58593692544131047267409907639, 7.56447096896047947573525794487, 7.954039143402871806607944942063, 8.987177181849458777860760329529
