| L(s) = 1 | − 2.23i·5-s − 5.73i·7-s − 14.4i·11-s − 22.5·13-s − 15.8i·17-s − 21.5i·19-s + (−14.6 + 17.7i)23-s − 5.00·25-s + 23.3·29-s + 52.1·31-s − 12.8·35-s − 29.3i·37-s − 69.3·41-s − 59.0i·43-s − 35.0·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 0.818i·7-s − 1.31i·11-s − 1.73·13-s − 0.935i·17-s − 1.13i·19-s + (−0.636 + 0.771i)23-s − 0.200·25-s + 0.804·29-s + 1.68·31-s − 0.366·35-s − 0.793i·37-s − 1.69·41-s − 1.37i·43-s − 0.746·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8799171400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8799171400\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (14.6 - 17.7i)T \) |
| good | 7 | \( 1 + 5.73iT - 49T^{2} \) |
| 11 | \( 1 + 14.4iT - 121T^{2} \) |
| 13 | \( 1 + 22.5T + 169T^{2} \) |
| 17 | \( 1 + 15.8iT - 289T^{2} \) |
| 19 | \( 1 + 21.5iT - 361T^{2} \) |
| 29 | \( 1 - 23.3T + 841T^{2} \) |
| 31 | \( 1 - 52.1T + 961T^{2} \) |
| 37 | \( 1 + 29.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 69.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 59.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 55.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 34.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 35.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 111. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 94.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 8.86T + 5.32e3T^{2} \) |
| 79 | \( 1 - 86.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 153. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 152. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 14.2iT - 9.40e3T^{2} \) |
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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
−7.902831317623859006635291919845, −6.98828429035602609061581485348, −6.56132634976745868732213283249, −5.23519034077044045764591868757, −5.01267853861370253499170789454, −3.99571483829362728010211487372, −3.09321759155288429064524139066, −2.24882002954977009769703265198, −0.824372069889394075467185796207, −0.21908074473728893107577747675,
1.61044669394827556121516279986, 2.38616296415069031966044530722, 3.08591985106587721668663372690, 4.42735428996785745652245070429, 4.76245374528699730993779539059, 5.86327475790794159799412651897, 6.48753574765426165387202321701, 7.21866045411521724979009530045, 8.023908613276369129996215870574, 8.504235563731002001526349571234
