L(s) = 1 | + (−1.09 − 1.34i)3-s + (−0.271 + 0.469i)5-s + (1.60 + 2.10i)7-s + (−0.599 + 2.93i)9-s + (−0.666 + 0.384i)11-s + (−2.96 + 1.71i)13-s + (0.926 − 0.150i)15-s + (−3.23 + 5.60i)17-s + (5.60 − 3.23i)19-s + (1.05 − 4.45i)21-s + (0.100 + 0.0579i)23-s + (2.35 + 4.07i)25-s + (4.59 − 2.41i)27-s + (4.40 + 2.54i)29-s + 4.63i·31-s + ⋯ |
L(s) = 1 | + (−0.632 − 0.774i)3-s + (−0.121 + 0.209i)5-s + (0.607 + 0.794i)7-s + (−0.199 + 0.979i)9-s + (−0.200 + 0.115i)11-s + (−0.822 + 0.474i)13-s + (0.239 − 0.0389i)15-s + (−0.784 + 1.35i)17-s + (1.28 − 0.742i)19-s + (0.231 − 0.972i)21-s + (0.0209 + 0.0120i)23-s + (0.470 + 0.815i)25-s + (0.885 − 0.465i)27-s + (0.817 + 0.471i)29-s + 0.832i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.890716 + 0.445358i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.890716 + 0.445358i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.09 + 1.34i)T \) |
| 7 | \( 1 + (-1.60 - 2.10i)T \) |
good | 5 | \( 1 + (0.271 - 0.469i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.666 - 0.384i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.96 - 1.71i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.23 - 5.60i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.60 + 3.23i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.100 - 0.0579i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.40 - 2.54i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.63iT - 31T^{2} \) |
| 37 | \( 1 + (-5.47 - 9.48i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.04 + 7.00i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.32 + 5.76i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.54T + 47T^{2} \) |
| 53 | \( 1 + (0.221 + 0.127i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 5.57iT - 61T^{2} \) |
| 67 | \( 1 + 3.28T + 67T^{2} \) |
| 71 | \( 1 - 5.67iT - 71T^{2} \) |
| 73 | \( 1 + (5.35 + 3.09i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 4.02T + 79T^{2} \) |
| 83 | \( 1 + (5.80 - 10.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.00 + 3.47i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.0 - 8.69i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.18885973978955778573021440443, −10.42494076158346280187414673617, −9.135911210992667117643883729067, −8.272690628669065394400283714151, −7.30341414459992383691808067567, −6.54948535791820967796415502165, −5.41185737959092313938887744648, −4.68689277350484836498663675438, −2.81139858186875936465973596935, −1.59891638019087857393595381486,
0.67521061778962391699496210073, 2.89363698594262383866529736236, 4.33678617070182797768240623597, 4.89737913011350059866757913334, 5.96478625942094837733030158115, 7.22058893137171496589475527780, 7.993737574704123516114018895496, 9.289651412746170326791789304823, 9.955591068669725269038760183503, 10.79399439386444333269050058863