| L(s) = 1 | + 1.92·2-s + 3.21·3-s + 1.70·4-s + 2.13·5-s + 6.19·6-s − 0.572·8-s + 7.36·9-s + 4.09·10-s + 1.29·11-s + 5.48·12-s − 4.97·13-s + 6.85·15-s − 4.50·16-s + 2.24·17-s + 14.1·18-s − 6.45·19-s + 3.62·20-s + 2.49·22-s + 3.94·23-s − 1.84·24-s − 0.462·25-s − 9.56·26-s + 14.0·27-s − 8.60·29-s + 13.1·30-s + 8.78·31-s − 7.52·32-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 1.85·3-s + 0.851·4-s + 0.952·5-s + 2.52·6-s − 0.202·8-s + 2.45·9-s + 1.29·10-s + 0.391·11-s + 1.58·12-s − 1.37·13-s + 1.77·15-s − 1.12·16-s + 0.545·17-s + 3.33·18-s − 1.48·19-s + 0.810·20-s + 0.532·22-s + 0.821·23-s − 0.376·24-s − 0.0925·25-s − 1.87·26-s + 2.70·27-s − 1.59·29-s + 2.40·30-s + 1.57·31-s − 1.33·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.323288174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.323288174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 3 | \( 1 - 3.21T + 3T^{2} \) |
| 5 | \( 1 - 2.13T + 5T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 - 8.78T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 43 | \( 1 + 7.55T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 - 3.99T + 53T^{2} \) |
| 59 | \( 1 + 6.02T + 59T^{2} \) |
| 61 | \( 1 - 6.14T + 61T^{2} \) |
| 67 | \( 1 + 9.88T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 + 2.27T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 6.78T + 83T^{2} \) |
| 89 | \( 1 + 3.71T + 89T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
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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
−9.152346210307810879515026802992, −8.486124481114046659612504050442, −7.47770383130662249659398398138, −6.77548527599452745452129125561, −5.86869734430732839753131619064, −4.82068655350991959407739040448, −4.18234274496558698685454893268, −3.24950998179607311403030082555, −2.50587668676622514313473279838, −1.85833923110188652895080107675,
1.85833923110188652895080107675, 2.50587668676622514313473279838, 3.24950998179607311403030082555, 4.18234274496558698685454893268, 4.82068655350991959407739040448, 5.86869734430732839753131619064, 6.77548527599452745452129125561, 7.47770383130662249659398398138, 8.486124481114046659612504050442, 9.152346210307810879515026802992
