| L(s) = 1 | + (2.41 + 1.64i)3-s + (−2.00 + 1.85i)5-s + (−1.01 + 1.75i)7-s + (2.01 + 5.13i)9-s + (0.515 − 0.646i)11-s + (3.02 + 0.931i)13-s + (−7.87 + 1.18i)15-s + (−2.50 − 2.32i)17-s + (1.08 − 2.75i)19-s + (−5.31 + 2.55i)21-s + (−1.21 − 0.182i)23-s + (0.183 − 2.44i)25-s + (−1.63 + 7.15i)27-s + (−7.00 + 4.77i)29-s + (0.399 + 5.32i)31-s + ⋯ |
| L(s) = 1 | + (1.39 + 0.949i)3-s + (−0.894 + 0.830i)5-s + (−0.382 + 0.661i)7-s + (0.671 + 1.71i)9-s + (0.155 − 0.194i)11-s + (0.837 + 0.258i)13-s + (−2.03 + 0.306i)15-s + (−0.607 − 0.563i)17-s + (0.248 − 0.632i)19-s + (−1.15 + 0.558i)21-s + (−0.252 − 0.0380i)23-s + (0.0366 − 0.489i)25-s + (−0.314 + 1.37i)27-s + (−1.30 + 0.886i)29-s + (0.0716 + 0.956i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.925854 + 1.64426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.925854 + 1.64426i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (-3.02 - 5.82i)T \) |
| good | 3 | \( 1 + (-2.41 - 1.64i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (2.00 - 1.85i)T + (0.373 - 4.98i)T^{2} \) |
| 7 | \( 1 + (1.01 - 1.75i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.515 + 0.646i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-3.02 - 0.931i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (2.50 + 2.32i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.08 + 2.75i)T + (-13.9 - 12.9i)T^{2} \) |
| 23 | \( 1 + (1.21 + 0.182i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (7.00 - 4.77i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-0.399 - 5.32i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (1.73 + 3.00i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.20 - 2.98i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (2.33 + 2.93i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-8.47 + 2.61i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.633 + 2.77i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-0.573 + 7.65i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-3.76 + 9.59i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (-11.9 + 1.80i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (-11.9 - 3.67i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (-2.34 + 4.05i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.94 - 2.68i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (1.70 + 1.16i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (7.51 - 9.42i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−10.92408707947754171743752594027, −9.592789148884862734916580156038, −9.107062337687200528741386606143, −8.355752515414029401912134247130, −7.45912964046499225874941572422, −6.52429722024586437959507894187, −5.06589683306253345563400269409, −3.85443359195784220285793996074, −3.32781413711856733382433385313, −2.37212913333878466840190024921,
0.877889410190841501502750357134, 2.20443395334616498466225314896, 3.75067069182498558937736667603, 4.03845943818650113678178894841, 5.86875907236927800711281946239, 7.01178239270046759838523529403, 7.74610944629228176395897641645, 8.332265767512920890747027776138, 9.011329559253333605724019887203, 9.934683203395520441405706797787
