| L(s) = 1 | + 3-s − 3.18·5-s − 1.10·7-s + 9-s − 11-s − 13-s − 3.18·15-s − 6.29·17-s − 3.74·19-s − 1.10·21-s − 3.26·23-s + 5.14·25-s + 27-s + 3.60·29-s + 0.336·31-s − 33-s + 3.52·35-s + 6.37·37-s − 39-s + 0.894·41-s − 8.85·43-s − 3.18·45-s + 8.21·47-s − 5.77·49-s − 6.29·51-s + 5.32·53-s + 3.18·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.42·5-s − 0.417·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s − 0.822·15-s − 1.52·17-s − 0.858·19-s − 0.241·21-s − 0.681·23-s + 1.02·25-s + 0.192·27-s + 0.668·29-s + 0.0604·31-s − 0.174·33-s + 0.595·35-s + 1.04·37-s − 0.160·39-s + 0.139·41-s − 1.35·43-s − 0.474·45-s + 1.19·47-s − 0.825·49-s − 0.880·51-s + 0.731·53-s + 0.429·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9097681147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9097681147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3.18T + 5T^{2} \) |
| 7 | \( 1 + 1.10T + 7T^{2} \) |
| 17 | \( 1 + 6.29T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 3.26T + 23T^{2} \) |
| 29 | \( 1 - 3.60T + 29T^{2} \) |
| 31 | \( 1 - 0.336T + 31T^{2} \) |
| 37 | \( 1 - 6.37T + 37T^{2} \) |
| 41 | \( 1 - 0.894T + 41T^{2} \) |
| 43 | \( 1 + 8.85T + 43T^{2} \) |
| 47 | \( 1 - 8.21T + 47T^{2} \) |
| 53 | \( 1 - 5.32T + 53T^{2} \) |
| 59 | \( 1 - 9.01T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 5.32T + 71T^{2} \) |
| 73 | \( 1 + 5.03T + 73T^{2} \) |
| 79 | \( 1 + 9.54T + 79T^{2} \) |
| 83 | \( 1 - 5.11T + 83T^{2} \) |
| 89 | \( 1 - 3.66T + 89T^{2} \) |
| 97 | \( 1 - 8.13T + 97T^{2} \) |
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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
−7.901117302951875777942755990484, −7.44871357579514465479321819018, −6.67230183754304370464556390726, −6.06432031103699448321016858177, −4.72023130585570917220570456472, −4.36320732904890544979701464020, −3.61572676144297510528591599137, −2.81200120341979243166919960193, −2.00844397431474141938390203981, −0.44786720538132208449908148316,
0.44786720538132208449908148316, 2.00844397431474141938390203981, 2.81200120341979243166919960193, 3.61572676144297510528591599137, 4.36320732904890544979701464020, 4.72023130585570917220570456472, 6.06432031103699448321016858177, 6.67230183754304370464556390726, 7.44871357579514465479321819018, 7.901117302951875777942755990484
