| L(s) = 1 | + 1.18·2-s − 2.80·3-s − 0.587·4-s − 2.45·5-s − 3.32·6-s − 2.24·7-s − 3.07·8-s + 4.84·9-s − 2.91·10-s − 0.270·11-s + 1.64·12-s + 5.10·13-s − 2.66·14-s + 6.87·15-s − 2.48·16-s + 3.58·17-s + 5.76·18-s + 0.250·19-s + 1.44·20-s + 6.27·21-s − 0.321·22-s − 23-s + 8.61·24-s + 1.01·25-s + 6.06·26-s − 5.17·27-s + 1.31·28-s + ⋯ |
| L(s) = 1 | + 0.840·2-s − 1.61·3-s − 0.293·4-s − 1.09·5-s − 1.35·6-s − 0.846·7-s − 1.08·8-s + 1.61·9-s − 0.921·10-s − 0.0815·11-s + 0.474·12-s + 1.41·13-s − 0.711·14-s + 1.77·15-s − 0.620·16-s + 0.868·17-s + 1.35·18-s + 0.0574·19-s + 0.322·20-s + 1.36·21-s − 0.0685·22-s − 0.208·23-s + 1.75·24-s + 0.203·25-s + 1.18·26-s − 0.996·27-s + 0.248·28-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
| \(L(1)\) |
\(\approx\) |
\(0.6541453507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6541453507\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.18T + 2T^{2} \) |
| 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 + 0.270T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 - 0.250T + 19T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 - 3.58T + 37T^{2} \) |
| 41 | \( 1 - 3.14T + 41T^{2} \) |
| 43 | \( 1 + 4.04T + 43T^{2} \) |
| 47 | \( 1 - 0.357T + 47T^{2} \) |
| 53 | \( 1 - 6.68T + 53T^{2} \) |
| 59 | \( 1 - 8.05T + 59T^{2} \) |
| 61 | \( 1 - 0.731T + 61T^{2} \) |
| 67 | \( 1 - 1.58T + 67T^{2} \) |
| 71 | \( 1 + 3.90T + 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−10.80964315666784335151200995663, −9.906532406293521028922585185138, −8.804580650538144097807045642615, −7.67522434887661964572531207213, −6.50732576807900914401777234512, −5.92674966866098475262318589834, −5.10004456140883364957070932272, −4.02862093382309318707959118095, −3.42881902781751840784875265482, −0.65207629588088117775402841103,
0.65207629588088117775402841103, 3.42881902781751840784875265482, 4.02862093382309318707959118095, 5.10004456140883364957070932272, 5.92674966866098475262318589834, 6.50732576807900914401777234512, 7.67522434887661964572531207213, 8.804580650538144097807045642615, 9.906532406293521028922585185138, 10.80964315666784335151200995663
