| L(s) = 1 | + (0.581 − 1.28i)2-s + (−1.32 − 1.50i)4-s + (−2.70 + 0.832i)8-s + 6.57·11-s + (−0.5 + 3.96i)16-s + (3.82 − 8.46i)22-s − 1.91·23-s + 5·25-s − 6.06i·29-s + (4.82 + 2.95i)32-s + 10.5·37-s − 12i·43-s + (−8.69 − 9.85i)44-s + (−1.11 + 2.46i)46-s + (2.90 − 6.44i)50-s + ⋯ |
| L(s) = 1 | + (0.411 − 0.911i)2-s + (−0.661 − 0.750i)4-s + (−0.955 + 0.294i)8-s + 1.98·11-s + (−0.125 + 0.992i)16-s + (0.815 − 1.80i)22-s − 0.399·23-s + 25-s − 1.12i·29-s + (0.852 + 0.522i)32-s + 1.73·37-s − 1.82i·43-s + (−1.31 − 1.48i)44-s + (−0.164 + 0.363i)46-s + (0.411 − 0.911i)50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.150981711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.150981711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.581 + 1.28i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 - 6.57T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 1.91T + 23T^{2} \) |
| 29 | \( 1 + 6.06iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14.5iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.179123229874955048498817478524, −8.673992120882978349621751476954, −7.46141682087182213837015012970, −6.37775744337147583751793034379, −5.88978845345405175028442626771, −4.59396540791073106202458533369, −4.06539189014804358345370952680, −3.11399202297402287835073104776, −1.94264850714508135166459931508, −0.867907152865859272279176720612,
1.20723827344751300427739586131, 2.92614850017699024705418321777, 3.91994937968984945359937857199, 4.55716618244743540308791449847, 5.59592070180733887223556195876, 6.50179343686056363373409898175, 6.86238133918261790600753668418, 7.88082215841577220635149648028, 8.703356829409532656086977717211, 9.269913763569990865207157418429
