| L(s) = 1 | + 3-s − 2.79i·5-s − i·7-s + 9-s + 1.93i·11-s + (−1.07 + 3.44i)13-s − 2.79i·15-s + 4.51·17-s + 6.15i·19-s − i·21-s + 1.79·23-s − 2.79·25-s + 27-s + 8.67·29-s − 4.87i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.24i·5-s − 0.377i·7-s + 0.333·9-s + 0.582i·11-s + (−0.298 + 0.954i)13-s − 0.720i·15-s + 1.09·17-s + 1.41i·19-s − 0.218i·21-s + 0.374·23-s − 0.558·25-s + 0.192·27-s + 1.61·29-s − 0.875i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.568070151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.568070151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (1.07 - 3.44i)T \) |
| good | 5 | \( 1 + 2.79iT - 5T^{2} \) |
| 11 | \( 1 - 1.93iT - 11T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 6.15iT - 19T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 - 8.67T + 29T^{2} \) |
| 31 | \( 1 + 4.87iT - 31T^{2} \) |
| 37 | \( 1 + 9.96iT - 37T^{2} \) |
| 41 | \( 1 - 5.88iT - 41T^{2} \) |
| 43 | \( 1 - 3.42T + 43T^{2} \) |
| 47 | \( 1 - 12.1iT - 47T^{2} \) |
| 53 | \( 1 + 0.361T + 53T^{2} \) |
| 59 | \( 1 + 5.51iT - 59T^{2} \) |
| 61 | \( 1 + 2.15T + 61T^{2} \) |
| 67 | \( 1 - 9.24iT - 67T^{2} \) |
| 71 | \( 1 - 6.06iT - 71T^{2} \) |
| 73 | \( 1 - 2.86iT - 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 3.67iT - 83T^{2} \) |
| 89 | \( 1 - 3.44iT - 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
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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
−8.218400627027326970022242213681, −7.82590910350135596523500525588, −7.04127771032387208560297887467, −6.09573720478159381136402474904, −5.25969591478137857422486059996, −4.40402850668747403293117587957, −4.01791913162456225251526610303, −2.84538246886871332451251011633, −1.73927077859868322350411666010, −0.977788208870383573784361447104,
0.855130711577169740974292419302, 2.34372345705825321879420004277, 3.11426770608689506766462843461, 3.31496649429698423777134171365, 4.74187256424366911107249944904, 5.42486524952214215719066003508, 6.43162444491638914546879801017, 6.92369673491746379116331710585, 7.69684441843758729368264184327, 8.370098209726659546673424296572
