| L(s) = 1 | + 4.72i·7-s − 3.93i·11-s + 3.46·13-s + 3.51i·17-s − 5.44i·19-s − 7.11i·23-s − 3.66i·29-s − 5.23·31-s − 0.414·37-s − 3.00·41-s + 5.34·43-s + 0.925i·47-s − 15.3·49-s − 0.233·53-s − 14.3i·59-s + ⋯ |
| L(s) = 1 | + 1.78i·7-s − 1.18i·11-s + 0.961·13-s + 0.852i·17-s − 1.24i·19-s − 1.48i·23-s − 0.681i·29-s − 0.940·31-s − 0.0681·37-s − 0.469·41-s + 0.815·43-s + 0.135i·47-s − 2.18·49-s − 0.0320·53-s − 1.87i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.304796513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304796513\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.72iT - 7T^{2} \) |
| 11 | \( 1 + 3.93iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.51iT - 17T^{2} \) |
| 19 | \( 1 + 5.44iT - 19T^{2} \) |
| 23 | \( 1 + 7.11iT - 23T^{2} \) |
| 29 | \( 1 + 3.66iT - 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 + 0.414T + 37T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 - 5.34T + 43T^{2} \) |
| 47 | \( 1 - 0.925iT - 47T^{2} \) |
| 53 | \( 1 + 0.233T + 53T^{2} \) |
| 59 | \( 1 + 14.3iT - 59T^{2} \) |
| 61 | \( 1 + 0.118iT - 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 + 0.563iT - 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 8.88T + 89T^{2} \) |
| 97 | \( 1 + 7.27iT - 97T^{2} \) |
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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
−8.118268552250863132666734558282, −6.89221486337316664972076504767, −6.12395077128913501431078206583, −5.86675083509295342763354624706, −5.06749494383976331966618639991, −4.16046735445094882377742464060, −3.19222013954390750904010492558, −2.59627041256326755561328545844, −1.68670604803841406389663256268, −0.32568516827759894899725355060,
1.14732122101178269342864263724, 1.71230058828455677105104471050, 3.13406671654597833591655151063, 3.86084675584375619605308259050, 4.31835779222052058345414489159, 5.23785526231151872963856884942, 5.99614488906453138405693833448, 6.97146726642781478895112073086, 7.36759903713014065091609361741, 7.77518407548297588473377818302
