| L(s) = 1 | + 3-s − 5-s − 1.80·7-s + 9-s + 5.02·11-s + 1.80·13-s − 15-s − 3.32·17-s − 6.64·19-s − 1.80·21-s + 3.32·23-s + 25-s + 27-s + 6.54·29-s + 31-s + 5.02·33-s + 1.80·35-s + 10.2·37-s + 1.80·39-s − 7.02·41-s + 3.67·43-s − 45-s − 10.9·47-s − 3.73·49-s − 3.32·51-s + 1.65·53-s − 5.02·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.682·7-s + 0.333·9-s + 1.51·11-s + 0.501·13-s − 0.258·15-s − 0.805·17-s − 1.52·19-s − 0.394·21-s + 0.692·23-s + 0.200·25-s + 0.192·27-s + 1.21·29-s + 0.179·31-s + 0.875·33-s + 0.305·35-s + 1.68·37-s + 0.289·39-s − 1.09·41-s + 0.559·43-s − 0.149·45-s − 1.60·47-s − 0.533·49-s − 0.465·51-s + 0.226·53-s − 0.678·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253067161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253067161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 - 5.02T + 11T^{2} \) |
| 13 | \( 1 - 1.80T + 13T^{2} \) |
| 17 | \( 1 + 3.32T + 17T^{2} \) |
| 19 | \( 1 + 6.64T + 19T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 7.02T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 1.65T + 53T^{2} \) |
| 59 | \( 1 - 8.54T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 2.44T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 - 7.80T + 73T^{2} \) |
| 79 | \( 1 - 7.96T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 3.80T + 97T^{2} \) |
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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
−8.076865273444481139271379341107, −7.03828718849902434381234136435, −6.50746329164085266777259353477, −6.19706689748894889805216710145, −4.78386991215907635836972561612, −4.21584154010321863438948333421, −3.58607168136397941856338405122, −2.80899573263421474706684359032, −1.83308172709289703058280887016, −0.74186666919694817558768396944,
0.74186666919694817558768396944, 1.83308172709289703058280887016, 2.80899573263421474706684359032, 3.58607168136397941856338405122, 4.21584154010321863438948333421, 4.78386991215907635836972561612, 6.19706689748894889805216710145, 6.50746329164085266777259353477, 7.03828718849902434381234136435, 8.076865273444481139271379341107
