| L(s) = 1 | + 2-s + 2.73·3-s + 4-s − 0.732·5-s + 2.73·6-s + 8-s + 4.46·9-s − 0.732·10-s − 0.732·11-s + 2.73·12-s + 4·13-s − 2·15-s + 16-s + 17-s + 4.46·18-s + 2·19-s − 0.732·20-s − 0.732·22-s + 6.92·23-s + 2.73·24-s − 4.46·25-s + 4·26-s + 3.99·27-s − 8.19·29-s − 2·30-s − 5.46·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.57·3-s + 0.5·4-s − 0.327·5-s + 1.11·6-s + 0.353·8-s + 1.48·9-s − 0.231·10-s − 0.220·11-s + 0.788·12-s + 1.10·13-s − 0.516·15-s + 0.250·16-s + 0.242·17-s + 1.05·18-s + 0.458·19-s − 0.163·20-s − 0.156·22-s + 1.44·23-s + 0.557·24-s − 0.892·25-s + 0.784·26-s + 0.769·27-s − 1.52·29-s − 0.365·30-s − 0.981·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.561646899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.561646899\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 + 0.732T + 5T^{2} \) |
| 11 | \( 1 + 0.732T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 - 0.535T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 6.53T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 - 7.46T + 83T^{2} \) |
| 89 | \( 1 + 9.85T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−9.167585061986404718257326499758, −8.592018186345220595038215436149, −7.64904170985318086712014469568, −7.29270267461326030104575549916, −6.07917153690965264091408618018, −5.14339943308635129936186442961, −3.91390349006024064341345060733, −3.52321809267554628469718244358, −2.59615771144791499620509396982, −1.49884629994253999817347760351,
1.49884629994253999817347760351, 2.59615771144791499620509396982, 3.52321809267554628469718244358, 3.91390349006024064341345060733, 5.14339943308635129936186442961, 6.07917153690965264091408618018, 7.29270267461326030104575549916, 7.64904170985318086712014469568, 8.592018186345220595038215436149, 9.167585061986404718257326499758
