| L(s) = 1 | + (4.68 − 2.70i)5-s + (6.12 − 3.38i)7-s + (−5.26 + 9.12i)11-s + 12.0i·13-s + (20.8 + 12.0i)17-s + (−30.9 + 17.8i)19-s + (20.6 + 35.7i)23-s + (2.11 − 3.65i)25-s + 28.6·29-s + (4.50 + 2.59i)31-s + (19.5 − 32.3i)35-s + (1.90 + 3.30i)37-s − 9.20i·41-s − 54.2·43-s + (14.3 − 8.27i)47-s + ⋯ |
| L(s) = 1 | + (0.936 − 0.540i)5-s + (0.875 − 0.483i)7-s + (−0.478 + 0.829i)11-s + 0.925i·13-s + (1.22 + 0.707i)17-s + (−1.62 + 0.939i)19-s + (0.898 + 1.55i)23-s + (0.0844 − 0.146i)25-s + 0.988·29-s + (0.145 + 0.0838i)31-s + (0.558 − 0.925i)35-s + (0.0516 + 0.0893i)37-s − 0.224i·41-s − 1.26·43-s + (0.305 − 0.176i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.422949932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.422949932\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-6.12 + 3.38i)T \) |
| good | 5 | \( 1 + (-4.68 + 2.70i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (5.26 - 9.12i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 12.0iT - 169T^{2} \) |
| 17 | \( 1 + (-20.8 - 12.0i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (30.9 - 17.8i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-20.6 - 35.7i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 28.6T + 841T^{2} \) |
| 31 | \( 1 + (-4.50 - 2.59i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-1.90 - 3.30i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 9.20iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 54.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-14.3 + 8.27i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-9.58 + 16.5i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-10.8 - 6.25i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-84.1 + 48.5i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-0.499 + 0.864i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 14.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (51.0 + 29.5i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (50.7 + 87.9i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 22.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-98.7 + 57.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 153. iT - 9.40e3T^{2} \) |
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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
−10.04314010317581693578796322283, −9.020154655600019257677451199443, −8.222752897419821593567172863845, −7.41824223477834493289950594464, −6.40458913956486946786844499289, −5.41712671623030343846508431934, −4.72794598218506327792291650492, −3.70487751588104750651346505947, −1.99417881012795296962604698172, −1.41294376928249043624990299592,
0.798638450848942317866512186349, 2.40614205599631251642050559476, 2.93603611663116376412064255443, 4.61870037642321732566374216407, 5.42135883883460755749343864815, 6.17574115589053717677353669925, 7.07600831944606429356292358751, 8.327089180739422868079713453358, 8.589959657833547005036558199599, 9.861228562369061314132757177873
