L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + (3 + 5.19i)11-s + 2·13-s + (2 − 3.46i)19-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s + 0.999·27-s + 6·29-s + (−4 − 6.92i)31-s + (3 − 5.19i)33-s + (−1 + 1.73i)37-s + (−1 − 1.73i)39-s + 12·41-s − 4·43-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.166 + 0.288i)9-s + (0.904 + 1.56i)11-s + 0.554·13-s + (0.458 − 0.794i)19-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + 0.192·27-s + 1.11·29-s + (−0.718 − 1.24i)31-s + (0.522 − 0.904i)33-s + (−0.164 + 0.284i)37-s + (−0.160 − 0.277i)39-s + 1.87·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43102 - 0.0908128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43102 - 0.0908128i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.87020691644171622528047026403, −9.675379819704747705888887866276, −9.055002276238311523165184370499, −7.87656267761473699156152974313, −6.97806787793626094328148046787, −6.37666893824194792476706273193, −5.06811282590913572284262078589, −4.17534176358488821742811419980, −2.62376870101628358339758927299, −1.23902767568951116073407019843,
1.11439675822313629821117992368, 3.16765464597929940935755513166, 3.92971438854184200707455607170, 5.28038697610206779737532920156, 6.05518776469725952841377631409, 6.97205489038106508013561606991, 8.367263437544741754394906904257, 8.870941719115200052223813108011, 9.903872468831631807239936662163, 10.79094653225598986408776129999