L(s) = 1 | + (−0.0309 + 0.0536i)5-s + (0.981 − 2.45i)7-s + (1.59 + 2.75i)11-s + (−0.252 − 0.437i)13-s + (0.554 − 0.960i)17-s + (−0.933 − 1.61i)19-s + (3.10 − 5.37i)23-s + (2.49 + 4.32i)25-s + (−2.39 + 4.15i)29-s + 2.53·31-s + (0.101 + 0.128i)35-s + (−4.26 − 7.38i)37-s + (4.94 + 8.56i)41-s + (3.95 − 6.85i)43-s + 6.58·47-s + ⋯ |
L(s) = 1 | + (−0.0138 + 0.0240i)5-s + (0.371 − 0.928i)7-s + (0.479 + 0.830i)11-s + (−0.0700 − 0.121i)13-s + (0.134 − 0.233i)17-s + (−0.214 − 0.370i)19-s + (0.646 − 1.12i)23-s + (0.499 + 0.865i)25-s + (−0.445 + 0.770i)29-s + 0.455·31-s + (0.0171 + 0.0217i)35-s + (−0.700 − 1.21i)37-s + (0.772 + 1.33i)41-s + (0.603 − 1.04i)43-s + 0.960·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.949344340\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.949344340\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.981 + 2.45i)T \) |
good | 5 | \( 1 + (0.0309 - 0.0536i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.59 - 2.75i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.252 + 0.437i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.554 + 0.960i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.933 + 1.61i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.10 + 5.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.39 - 4.15i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 + (4.26 + 7.38i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.94 - 8.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.95 + 6.85i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.58T + 47T^{2} \) |
| 53 | \( 1 + (-1.58 + 2.74i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 9.01T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 3.33T + 67T^{2} \) |
| 71 | \( 1 - 2.25T + 71T^{2} \) |
| 73 | \( 1 + (-2.07 + 3.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 2.97T + 79T^{2} \) |
| 83 | \( 1 + (-2.17 + 3.76i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.30 - 7.44i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.27 - 5.67i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.753772802091611086992920226577, −7.69933692306248319381927424296, −7.15140686685614881794384620544, −6.61116512880476685368293474497, −5.45124231256997153557281193074, −4.66130873464405901858500257307, −4.03356443758291003583925733175, −3.00008964498267561224187929831, −1.85478120030939707437208809223, −0.74003582493394345573528459454,
1.06411704404200740913243688375, 2.21963999400323508063114466859, 3.15231562349684019750063945475, 4.09313481975711395448661430979, 5.03107222577758386661111603312, 5.85140814781197938283019003755, 6.34129220179858774446102428013, 7.42065644221988520959133585947, 8.147968782880034593313115136346, 8.851996125953842252871388829302