| L(s) = 1 | + (1.64 + 1.51i)5-s + 2.05·7-s − 5.60·11-s + 2.63i·13-s − 2.12i·17-s + 4.87i·19-s + (−4.47 − 1.71i)23-s + (0.412 + 4.98i)25-s − 1.54i·29-s − 6.05·31-s + (3.38 + 3.11i)35-s + 0.289·37-s − 2.93i·41-s − 12.2·43-s + 0.475·47-s + ⋯ |
| L(s) = 1 | + (0.735 + 0.677i)5-s + 0.777·7-s − 1.68·11-s + 0.729i·13-s − 0.516i·17-s + 1.11i·19-s + (−0.933 − 0.358i)23-s + (0.0825 + 0.996i)25-s − 0.287i·29-s − 1.08·31-s + (0.571 + 0.526i)35-s + 0.0475·37-s − 0.459i·41-s − 1.86·43-s + 0.0693·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6939505610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6939505610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.64 - 1.51i)T \) |
| 23 | \( 1 + (4.47 + 1.71i)T \) |
| good | 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 - 2.63iT - 13T^{2} \) |
| 17 | \( 1 + 2.12iT - 17T^{2} \) |
| 19 | \( 1 - 4.87iT - 19T^{2} \) |
| 29 | \( 1 + 1.54iT - 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 - 0.289T + 37T^{2} \) |
| 41 | \( 1 + 2.93iT - 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 0.475T + 47T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 - 0.760iT - 61T^{2} \) |
| 67 | \( 1 - 0.897T + 67T^{2} \) |
| 71 | \( 1 - 1.57iT - 71T^{2} \) |
| 73 | \( 1 - 3.69iT - 73T^{2} \) |
| 79 | \( 1 - 13.5iT - 79T^{2} \) |
| 83 | \( 1 + 6.38iT - 83T^{2} \) |
| 89 | \( 1 + 4.36T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−8.592374888697143662289406197561, −8.035667575647144449078786770093, −7.32806691371219313831874952339, −6.61637781932403099420908415849, −5.62830250177172622077259675362, −5.27695124715040814038570042425, −4.27561112803322179596489164264, −3.25058451319981490562781495600, −2.28486370126772708536009038842, −1.71222299054715631712700059609,
0.17281217036234896799412403150, 1.55438768473389243098836917429, 2.34896465627542122846011543505, 3.30124845018116326671293986811, 4.58066178283339951865170815470, 5.13978367038075626262782721078, 5.61492759447958797449247638370, 6.50599333698346695138074195541, 7.62183173005014673912354580122, 8.048470061169379928459690461743
