| L(s) = 1 | − 1.11·5-s + (1.37 + 2.26i)7-s − 0.812·11-s − 2·13-s − 3.55i·17-s + 4.52i·19-s − 3.63i·23-s − 3.75·25-s + 8.95i·29-s + 5.08·31-s + (−1.53 − 2.52i)35-s + 0.718i·37-s − 10.4i·41-s + 1.11·43-s − 8.55·47-s + ⋯ |
| L(s) = 1 | − 0.498·5-s + (0.518 + 0.854i)7-s − 0.245·11-s − 0.554·13-s − 0.862i·17-s + 1.03i·19-s − 0.758i·23-s − 0.751·25-s + 1.66i·29-s + 0.914·31-s + (−0.258 − 0.426i)35-s + 0.118i·37-s − 1.63i·41-s + 0.170·43-s − 1.24·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08378427691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08378427691\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.37 - 2.26i)T \) |
| good | 5 | \( 1 + 1.11T + 5T^{2} \) |
| 11 | \( 1 + 0.812T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 3.55iT - 17T^{2} \) |
| 19 | \( 1 - 4.52iT - 19T^{2} \) |
| 23 | \( 1 + 3.63iT - 23T^{2} \) |
| 29 | \( 1 - 8.95iT - 29T^{2} \) |
| 31 | \( 1 - 5.08T + 31T^{2} \) |
| 37 | \( 1 - 0.718iT - 37T^{2} \) |
| 41 | \( 1 + 10.4iT - 41T^{2} \) |
| 43 | \( 1 - 1.11T + 43T^{2} \) |
| 47 | \( 1 + 8.55T + 47T^{2} \) |
| 53 | \( 1 - 5.08iT - 53T^{2} \) |
| 59 | \( 1 + 2.23iT - 59T^{2} \) |
| 61 | \( 1 + 5.27T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 5.87iT - 71T^{2} \) |
| 73 | \( 1 + 3.86iT - 73T^{2} \) |
| 79 | \( 1 - 1.70iT - 79T^{2} \) |
| 83 | \( 1 + 7.27iT - 83T^{2} \) |
| 89 | \( 1 - 3.37iT - 89T^{2} \) |
| 97 | \( 1 - 8.55iT - 97T^{2} \) |
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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
−8.824725324008549992696806537853, −8.027185036137846545813441570256, −7.56086962902723498698326312270, −6.67580428286565439006627991796, −5.79557567764469534251805692278, −5.07651306984101990123836465164, −4.44220314933277957201827482072, −3.35868789466849332203181193680, −2.54738907853975022906155031966, −1.55367779793375966101535866063,
0.02414805292597817478619217031, 1.29410827722097238949445818143, 2.43421425898076652932233304201, 3.46602333582022753508774123396, 4.34548621874370645473108242033, 4.78611781505950119697783418098, 5.87220946418972613768617318040, 6.63979327224829214292583240873, 7.52794236174387555227930459213, 7.903501092519157001758571606910
