| L(s) = 1 | − 0.405·3-s − 5.36·5-s + 18.2·7-s − 26.8·9-s + 45.3·11-s + 82.2·13-s + 2.17·15-s − 5.01·17-s − 25.6·19-s − 7.39·21-s + 23·23-s − 96.2·25-s + 21.8·27-s − 43.7·29-s + 336.·31-s − 18.4·33-s − 97.6·35-s + 246.·37-s − 33.3·39-s − 488.·41-s + 505.·43-s + 143.·45-s − 444.·47-s − 11.2·49-s + 2.03·51-s − 549.·53-s − 243.·55-s + ⋯ |
| L(s) = 1 | − 0.0780·3-s − 0.479·5-s + 0.983·7-s − 0.993·9-s + 1.24·11-s + 1.75·13-s + 0.0374·15-s − 0.0715·17-s − 0.310·19-s − 0.0768·21-s + 0.208·23-s − 0.769·25-s + 0.155·27-s − 0.280·29-s + 1.94·31-s − 0.0970·33-s − 0.471·35-s + 1.09·37-s − 0.136·39-s − 1.86·41-s + 1.79·43-s + 0.476·45-s − 1.38·47-s − 0.0326·49-s + 0.00558·51-s − 1.42·53-s − 0.596·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.363175476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.363175476\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 0.405T + 27T^{2} \) |
| 5 | \( 1 + 5.36T + 125T^{2} \) |
| 7 | \( 1 - 18.2T + 343T^{2} \) |
| 11 | \( 1 - 45.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.01T + 4.91e3T^{2} \) |
| 19 | \( 1 + 25.6T + 6.85e3T^{2} \) |
| 29 | \( 1 + 43.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 336.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 246.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 488.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 505.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 444.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 549.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 144.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 360.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 142.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 206.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 917.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 320.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 134.T + 9.12e5T^{2} \) |
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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
−8.856123002532946759888395582652, −8.394706321536704180078117872120, −7.77972452861556240985555004157, −6.44089652696802742643442200356, −6.07283462517296250855576113434, −4.85987588093021200043826591492, −4.03269682935774204073371397074, −3.19493439463206085987331740582, −1.76935253926689321816642774912, −0.800444164613185846375939405987,
0.800444164613185846375939405987, 1.76935253926689321816642774912, 3.19493439463206085987331740582, 4.03269682935774204073371397074, 4.85987588093021200043826591492, 6.07283462517296250855576113434, 6.44089652696802742643442200356, 7.77972452861556240985555004157, 8.394706321536704180078117872120, 8.856123002532946759888395582652
