L(s) = 1 | − 2.62·5-s − 0.432·7-s + 5.01·11-s − 1.56i·13-s − 19.8i·17-s + 24.1i·19-s − 3.07i·23-s − 18.1·25-s − 34.4·29-s + 43.4·31-s + 1.13·35-s − 52.9i·37-s − 56.7i·41-s + 27.6i·43-s − 83.7i·47-s + ⋯ |
L(s) = 1 | − 0.524·5-s − 0.0617·7-s + 0.455·11-s − 0.120i·13-s − 1.16i·17-s + 1.27i·19-s − 0.133i·23-s − 0.725·25-s − 1.18·29-s + 1.40·31-s + 0.0323·35-s − 1.43i·37-s − 1.38i·41-s + 0.643i·43-s − 1.78i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9910056621\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9910056621\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.62T + 25T^{2} \) |
| 7 | \( 1 + 0.432T + 49T^{2} \) |
| 11 | \( 1 - 5.01T + 121T^{2} \) |
| 13 | \( 1 + 1.56iT - 169T^{2} \) |
| 17 | \( 1 + 19.8iT - 289T^{2} \) |
| 19 | \( 1 - 24.1iT - 361T^{2} \) |
| 23 | \( 1 + 3.07iT - 529T^{2} \) |
| 29 | \( 1 + 34.4T + 841T^{2} \) |
| 31 | \( 1 - 43.4T + 961T^{2} \) |
| 37 | \( 1 + 52.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 56.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 27.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 83.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 41.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 74.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 28.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 33.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 53.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 51.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 76.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 68.9T + 9.40e3T^{2} \) |
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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
−9.695756503009130333931025446809, −8.910715342753473478062448182330, −7.914112176272117502670558597585, −7.29183547203701587437869299227, −6.23449446189124684954903824485, −5.30912067049114530773637233370, −4.16921538207397100092054440685, −3.35311460403231084647447386841, −1.94757135261762573288396116563, −0.34300000207301719277076094333,
1.32776100244021745356465117732, 2.80576984171132211887016945051, 3.93512738098019111752013660053, 4.73107296987061237350594960309, 5.99666190029063484807295614751, 6.74477611425716132082639062791, 7.76373053651081296781971612750, 8.462839657042549274289708299780, 9.391816765454986133642926912315, 10.15876455465654943556235398587