| L(s) = 1 | + 2-s + 0.484·3-s + 4-s + 3.12·5-s + 0.484·6-s − 7-s + 8-s − 2.76·9-s + 3.12·10-s + 4.64·11-s + 0.484·12-s − 14-s + 1.51·15-s + 16-s − 2.48·17-s − 2.76·18-s + 1.03·19-s + 3.12·20-s − 0.484·21-s + 4.64·22-s − 4.24·23-s + 0.484·24-s + 4.76·25-s − 2.79·27-s − 28-s − 1.12·29-s + 1.51·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.279·3-s + 0.5·4-s + 1.39·5-s + 0.197·6-s − 0.377·7-s + 0.353·8-s − 0.921·9-s + 0.988·10-s + 1.39·11-s + 0.139·12-s − 0.267·14-s + 0.391·15-s + 0.250·16-s − 0.602·17-s − 0.651·18-s + 0.236·19-s + 0.698·20-s − 0.105·21-s + 0.989·22-s − 0.886·23-s + 0.0989·24-s + 0.952·25-s − 0.537·27-s − 0.188·28-s − 0.208·29-s + 0.276·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.845378639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.845378639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 0.484T + 3T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 2.48T + 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 8.88T + 59T^{2} \) |
| 61 | \( 1 + 1.84T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 9.03T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 7.60T + 83T^{2} \) |
| 89 | \( 1 - 2.23T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−10.81444790505201579813622560969, −9.671194476078286043008974889804, −9.229426680917085371189358931732, −8.129630328776485933218086692755, −6.64596758441776910553999610122, −6.19769143318751476306815788592, −5.30799897326736813434100659852, −4.00608475745588460458165999242, −2.84344174822318155315235928310, −1.75206455187883530280281519592,
1.75206455187883530280281519592, 2.84344174822318155315235928310, 4.00608475745588460458165999242, 5.30799897326736813434100659852, 6.19769143318751476306815788592, 6.64596758441776910553999610122, 8.129630328776485933218086692755, 9.229426680917085371189358931732, 9.671194476078286043008974889804, 10.81444790505201579813622560969
