| L(s) = 1 | + 2.44·2-s + 3.99·4-s + 2.44·5-s + 4.89·8-s + 5.99·10-s + 4.89·11-s − 4·13-s + 3.99·16-s − 2.44·17-s − 19-s + 9.79·20-s + 11.9·22-s + 2.44·23-s + 0.999·25-s − 9.79·26-s − 7.34·29-s − 7·31-s − 5.99·34-s + 8·37-s − 2.44·38-s + 11.9·40-s + 7.34·41-s − 43-s + 19.5·44-s + 5.99·46-s + 2.44·47-s + 2.44·50-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 1.99·4-s + 1.09·5-s + 1.73·8-s + 1.89·10-s + 1.47·11-s − 1.10·13-s + 0.999·16-s − 0.594·17-s − 0.229·19-s + 2.19·20-s + 2.55·22-s + 0.510·23-s + 0.199·25-s − 1.92·26-s − 1.36·29-s − 1.25·31-s − 1.02·34-s + 1.31·37-s − 0.397·38-s + 1.89·40-s + 1.14·41-s − 0.152·43-s + 2.95·44-s + 0.884·46-s + 0.357·47-s + 0.346·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.480410873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.480410873\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 + 7.34T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 2.44T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 2.44T + 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−9.452144102853413527653525586633, −9.228577521888416344611621064031, −7.56076006955269034205658484004, −6.77713073850347960008532016184, −6.09688137784502721804176671179, −5.43103351972531468928767838787, −4.51509961686787975786451916627, −3.75018725417803188042230433155, −2.55380775165440784502363522246, −1.75514751078195191206685243181,
1.75514751078195191206685243181, 2.55380775165440784502363522246, 3.75018725417803188042230433155, 4.51509961686787975786451916627, 5.43103351972531468928767838787, 6.09688137784502721804176671179, 6.77713073850347960008532016184, 7.56076006955269034205658484004, 9.228577521888416344611621064031, 9.452144102853413527653525586633
