| L(s) = 1 | + 1.41·2-s + (1.52 + 2.58i)3-s + 2.00·4-s + (2.16 + 3.65i)6-s + 7.48i·7-s + 2.82·8-s + (−4.32 + 7.89i)9-s − 8.48i·11-s + (3.05 + 5.16i)12-s − 10i·13-s + 10.5i·14-s + 4.00·16-s + 30.3·17-s + (−6.11 + 11.1i)18-s − 26.9·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (0.509 + 0.860i)3-s + 0.500·4-s + (0.360 + 0.608i)6-s + 1.06i·7-s + 0.353·8-s + (−0.480 + 0.876i)9-s − 0.771i·11-s + (0.254 + 0.430i)12-s − 0.769i·13-s + 0.756i·14-s + 0.250·16-s + 1.78·17-s + (−0.339 + 0.620i)18-s − 1.41·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.16121 + 1.17847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16121 + 1.17847i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + (-1.52 - 2.58i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7.48iT - 49T^{2} \) |
| 11 | \( 1 + 8.48iT - 121T^{2} \) |
| 13 | \( 1 + 10iT - 169T^{2} \) |
| 17 | \( 1 - 30.3T + 289T^{2} \) |
| 19 | \( 1 + 26.9T + 361T^{2} \) |
| 23 | \( 1 - 9.17T + 529T^{2} \) |
| 29 | \( 1 + 26.8iT - 841T^{2} \) |
| 31 | \( 1 - 8T + 961T^{2} \) |
| 37 | \( 1 + 15.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 47.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 45.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 30.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 24.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 53.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 15.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 87.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 46.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 26.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 60.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 36.0iT - 9.40e3T^{2} \) |
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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
−12.98314935262949686336331196059, −12.03560688224170915022710444779, −10.92652949290726808765235874023, −9.993352591803428770687886484087, −8.749427492223122272097870633545, −7.896578276129167112954996531661, −5.98664784951455197102931214773, −5.21076881641888786069541884885, −3.69558346587960620401120730404, −2.58926661467084293340603813935,
1.55235360461407244243748556206, 3.28628954233851860633489864972, 4.59682451486433025647897038491, 6.32184853307430497579382756250, 7.18726871974569952408570515948, 8.089423692445458428968936679018, 9.588874636595068378878493523961, 10.74665530356542888354935279185, 12.00337832099193586772924188548, 12.72930667842318176411271279812
