| L(s) = 1 | + 8.89·3-s − 5·5-s + 20.4·7-s + 52.1·9-s + 61.3·11-s − 45.1·13-s − 44.4·15-s − 115.·17-s + 64.8·19-s + 182.·21-s − 6.11·23-s + 25·25-s + 224.·27-s + 224.·29-s − 58.6·31-s + 546.·33-s − 102.·35-s + 99.6·37-s − 402.·39-s − 145.·41-s − 6.73·43-s − 260.·45-s + 203.·47-s + 77.0·49-s − 1.02e3·51-s − 275.·53-s − 306.·55-s + ⋯ |
| L(s) = 1 | + 1.71·3-s − 0.447·5-s + 1.10·7-s + 1.93·9-s + 1.68·11-s − 0.964·13-s − 0.765·15-s − 1.64·17-s + 0.782·19-s + 1.89·21-s − 0.0554·23-s + 0.200·25-s + 1.59·27-s + 1.43·29-s − 0.339·31-s + 2.88·33-s − 0.494·35-s + 0.442·37-s − 1.65·39-s − 0.555·41-s − 0.0238·43-s − 0.864·45-s + 0.632·47-s + 0.224·49-s − 2.82·51-s − 0.714·53-s − 0.752·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.656652842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.656652842\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 3 | \( 1 - 8.89T + 27T^{2} \) |
| 7 | \( 1 - 20.4T + 343T^{2} \) |
| 11 | \( 1 - 61.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 6.11T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 58.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 99.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 6.73T + 7.95e4T^{2} \) |
| 47 | \( 1 - 203.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 275.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 790.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 141.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 826.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 154.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 414.T + 9.12e5T^{2} \) |
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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
−11.35074646340692991093220602664, −9.988568235616591623445927844562, −9.018944419770304718604626988426, −8.530670409625663267842511857835, −7.54799724843880417158256843138, −6.75361863257192645260788845205, −4.69437909248748009493025920604, −3.97207275760537128065400796402, −2.63772794902465869225771332254, −1.47613412646133334359214698269,
1.47613412646133334359214698269, 2.63772794902465869225771332254, 3.97207275760537128065400796402, 4.69437909248748009493025920604, 6.75361863257192645260788845205, 7.54799724843880417158256843138, 8.530670409625663267842511857835, 9.018944419770304718604626988426, 9.988568235616591623445927844562, 11.35074646340692991093220602664
