| L(s) = 1 | − i·2-s + 3-s − 4-s − i·5-s − i·6-s − 4.44·7-s + i·8-s + 9-s − 10-s − 2.44·11-s − 12-s − 0.440i·13-s + 4.44i·14-s − i·15-s + 16-s + 4.83i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s − 1.67·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.735·11-s − 0.288·12-s − 0.122i·13-s + 1.18i·14-s − 0.258i·15-s + 0.250·16-s + 1.17i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0326 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0326 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4269868788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4269868788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 37 | \( 1 + (0.198 - 6.07i)T \) |
| good | 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 0.440iT - 13T^{2} \) |
| 17 | \( 1 - 4.83iT - 17T^{2} \) |
| 19 | \( 1 - 0.440iT - 19T^{2} \) |
| 23 | \( 1 - 2.83iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 5.27iT - 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 0.396iT - 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 2.39iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 9.67T + 71T^{2} \) |
| 73 | \( 1 - 5.71T + 73T^{2} \) |
| 79 | \( 1 + 13.2iT - 79T^{2} \) |
| 83 | \( 1 + 7.71T + 83T^{2} \) |
| 89 | \( 1 + 15.3iT - 89T^{2} \) |
| 97 | \( 1 - 2.88iT - 97T^{2} \) |
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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
−10.14534018266567159340487372193, −9.295820885113401048653747054865, −8.628086987810301166379856845241, −7.78450579861553867267090221853, −6.67558134653437033217405456223, −5.77313572608271397040835871856, −4.65028642070425191317479203836, −3.48534724967726944665300173346, −3.00128140812606589667031366775, −1.60928711298272813010075349699,
0.16752176211543677570180018239, 2.55380178145259060831776332519, 3.27456400561402474595884393913, 4.35748155591622308533367233526, 5.57015719843406120080524538559, 6.45914230597440256840118688956, 7.10068981641414724842849812553, 7.84238938672044003310368886005, 8.858243249905826382408760401069, 9.642282284411137734600918924071
