| L(s) = 1 | + 19i·7-s − 22·11-s + i·13-s − 58i·17-s + 53·19-s − 58i·23-s + 22·29-s − 35·31-s + 270i·37-s + 468·41-s − 431i·43-s − 230i·47-s − 18·49-s + 446·59-s + 127·61-s + ⋯ |
| L(s) = 1 | + 1.02i·7-s − 0.603·11-s + 0.0213i·13-s − 0.827i·17-s + 0.639·19-s − 0.525i·23-s + 0.140·29-s − 0.202·31-s + 1.19i·37-s + 1.78·41-s − 1.52i·43-s − 0.713i·47-s − 0.0524·49-s + 0.984·59-s + 0.266·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.811979851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.811979851\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 19iT - 343T^{2} \) |
| 11 | \( 1 + 22T + 1.33e3T^{2} \) |
| 13 | \( 1 - iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 58iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 53T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 22T + 2.43e4T^{2} \) |
| 31 | \( 1 + 35T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 468T + 6.89e4T^{2} \) |
| 43 | \( 1 + 431iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 230iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 - 446T + 2.05e5T^{2} \) |
| 61 | \( 1 - 127T + 2.26e5T^{2} \) |
| 67 | \( 1 - 811iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 36T + 3.57e5T^{2} \) |
| 73 | \( 1 - 522iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.36e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.13e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 144T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.07e3iT - 9.12e5T^{2} \) |
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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
−8.978953864563239822186012059711, −8.386410561804469857769750618171, −7.47625665065262454068042498570, −6.71394656566363103196436034771, −5.62838586415054428676283680000, −5.21112062651992409561914800821, −4.10577686294779830313008645946, −2.88938623916626015743178619688, −2.28651331479010322288551122849, −0.840988129822077940071024306653,
0.49358286890047522587309732690, 1.56349116405121224044442575562, 2.82882624961970486927670791969, 3.80897897010173835143947235725, 4.56214095566332762806990356334, 5.59283556204901695296822275329, 6.34742313617586669267331525610, 7.54008219943278856057609697814, 7.64580490535863809511218551226, 8.797085239517514994844994507050
