| L(s) = 1 | + 1.88·2-s + 0.985·3-s + 1.56·4-s + 3.37·5-s + 1.86·6-s + 1.01·7-s − 0.818·8-s − 2.02·9-s + 6.37·10-s − 3.49·11-s + 1.54·12-s + 6.75·13-s + 1.92·14-s + 3.32·15-s − 4.67·16-s + 0.401·17-s − 3.83·18-s − 4.47·19-s + 5.29·20-s + 1.00·21-s − 6.60·22-s + 23-s − 0.806·24-s + 6.41·25-s + 12.7·26-s − 4.95·27-s + 1.59·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.568·3-s + 0.783·4-s + 1.51·5-s + 0.759·6-s + 0.384·7-s − 0.289·8-s − 0.676·9-s + 2.01·10-s − 1.05·11-s + 0.445·12-s + 1.87·13-s + 0.514·14-s + 0.859·15-s − 1.16·16-s + 0.0973·17-s − 0.903·18-s − 1.02·19-s + 1.18·20-s + 0.218·21-s − 1.40·22-s + 0.208·23-s − 0.164·24-s + 1.28·25-s + 2.50·26-s − 0.953·27-s + 0.301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.989158379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.989158379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 3 | \( 1 - 0.985T + 3T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 - 6.75T + 13T^{2} \) |
| 17 | \( 1 - 0.401T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 31 | \( 1 + 9.89T + 31T^{2} \) |
| 37 | \( 1 - 7.54T + 37T^{2} \) |
| 41 | \( 1 - 9.45T + 41T^{2} \) |
| 43 | \( 1 + 0.331T + 43T^{2} \) |
| 47 | \( 1 + 3.53T + 47T^{2} \) |
| 53 | \( 1 + 6.65T + 53T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 - 5.30T + 67T^{2} \) |
| 71 | \( 1 - 8.86T + 71T^{2} \) |
| 73 | \( 1 + 2.05T + 73T^{2} \) |
| 79 | \( 1 + 1.29T + 79T^{2} \) |
| 83 | \( 1 + 2.18T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−10.89990714354668053527977237437, −9.528323318227567868727116239790, −8.839820527975051404017962252381, −7.989737150482857970728681614620, −6.41341628037971020260322312925, −5.83084978897632184148886691745, −5.20611517309137580373834399907, −3.92207040402232253387865405916, −2.86511940333301402770566206980, −1.95710475289815049042525174035,
1.95710475289815049042525174035, 2.86511940333301402770566206980, 3.92207040402232253387865405916, 5.20611517309137580373834399907, 5.83084978897632184148886691745, 6.41341628037971020260322312925, 7.989737150482857970728681614620, 8.839820527975051404017962252381, 9.528323318227567868727116239790, 10.89990714354668053527977237437
