L(s) = 1 | − 8.48i·5-s + 7i·7-s + 8.48·11-s + 19.0i·13-s − 14.6·17-s − 8.66·19-s − 14.6i·23-s − 46.9·25-s − 50.9i·29-s − 10i·31-s + 59.3·35-s − 19.0i·37-s − 58.7·41-s + 41.5·43-s − 73.4i·47-s + ⋯ |
L(s) = 1 | − 1.69i·5-s + i·7-s + 0.771·11-s + 1.46i·13-s − 0.864·17-s − 0.455·19-s − 0.638i·23-s − 1.87·25-s − 1.75i·29-s − 0.322i·31-s + 1.69·35-s − 0.514i·37-s − 1.43·41-s + 0.966·43-s − 1.56i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6955823076\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6955823076\) |
\(L(2)\) |
|
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 | \( 1 + 8.48iT - 25T^{2} \) |
| 7 | \( 1 - 7iT - 49T^{2} \) |
| 11 | \( 1 - 8.48T + 121T^{2} \) |
| 13 | \( 1 - 19.0iT - 169T^{2} \) |
| 17 | \( 1 + 14.6T + 289T^{2} \) |
| 19 | \( 1 + 8.66T + 361T^{2} \) |
| 23 | \( 1 + 14.6iT - 529T^{2} \) |
| 29 | \( 1 + 50.9iT - 841T^{2} \) |
| 31 | \( 1 + 10iT - 961T^{2} \) |
| 37 | \( 1 + 19.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 58.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 41.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 73.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 25.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 19.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 116.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 117. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 71T + 5.32e3T^{2} \) |
| 79 | \( 1 + 151iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 135.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 161.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 25T + 9.40e3T^{2} \) |
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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
−8.742565792437904835187246200232, −8.462770860801510593953336711048, −7.14142795900223301741607562958, −6.20681002519973579511243026027, −5.53252692575068334953824130690, −4.36532406247278189159555331878, −4.22240078453075033970633743199, −2.34646427030698464644893233113, −1.58411404395506051792494799540, −0.17894393778318282928850578274,
1.41692900422705889589498668637, 2.84961421672070501153407091319, 3.42933872523375038331995076719, 4.35379692003584447989242360620, 5.60481315753453305025864551871, 6.55442365440637393038080547111, 7.03462361181466454889461036491, 7.68508084828217157420081052168, 8.656660145267345785099154873987, 9.708276550305351042761810550505