| L(s) = 1 | − 2.41i·2-s − 3.82·4-s + 2.41i·7-s + 4.41i·8-s − 6.41·11-s − i·13-s + 5.82·14-s + 2.99·16-s + 3.82i·17-s + 3.65·19-s + 15.4i·22-s − 3.17i·23-s − 2.41·26-s − 9.24i·28-s − 0.171·29-s + ⋯ |
| L(s) = 1 | − 1.70i·2-s − 1.91·4-s + 0.912i·7-s + 1.56i·8-s − 1.93·11-s − 0.277i·13-s + 1.55·14-s + 0.749·16-s + 0.928i·17-s + 0.838·19-s + 3.30i·22-s − 0.661i·23-s − 0.473·26-s − 1.74i·28-s − 0.0318·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.202585783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.202585783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 + 2.41iT - 2T^{2} \) |
| 7 | \( 1 - 2.41iT - 7T^{2} \) |
| 11 | \( 1 + 6.41T + 11T^{2} \) |
| 17 | \( 1 - 3.82iT - 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 + 3.17iT - 23T^{2} \) |
| 29 | \( 1 + 0.171T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 0.828iT - 43T^{2} \) |
| 47 | \( 1 + 3.58iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 2.75iT - 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 6.89iT - 83T^{2} \) |
| 89 | \( 1 + 8.48T + 89T^{2} \) |
| 97 | \( 1 + 0.828iT - 97T^{2} \) |
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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
−8.586103189439635153102392550469, −8.202582683224961598146106150919, −7.17861278600194131431624067180, −5.76801009423955381661991631722, −5.31749704677881817972305373803, −4.39038684146317382678329003968, −3.38739930268384852964131204809, −2.58613182662098985776943457755, −2.08857770246255187522710336611, −0.62574131222291821066748274744,
0.68528087287477569589223603097, 2.56851517990264228619921251063, 3.70963507536633233598538881534, 4.83190338911288352018573193221, 5.15527496094235511277281412854, 6.01156856157052536703001198748, 6.92822413805576765790521697281, 7.54741624523538945764605785839, 7.82909666470777322829615592069, 8.694292126680167947652987933539
