| L(s) = 1 | − 2·2-s + 9.05·3-s + 4·4-s + 3.66·5-s − 18.1·6-s + 10.1·7-s − 8·8-s + 54.9·9-s − 7.33·10-s + 15.0·11-s + 36.2·12-s + 62.4·13-s − 20.2·14-s + 33.2·15-s + 16·16-s + 104.·17-s − 109.·18-s + 113.·19-s + 14.6·20-s + 91.6·21-s − 30.0·22-s − 72.4·24-s − 111.·25-s − 124.·26-s + 253.·27-s + 40.4·28-s + 3.57·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.74·3-s + 0.5·4-s + 0.328·5-s − 1.23·6-s + 0.546·7-s − 0.353·8-s + 2.03·9-s − 0.231·10-s + 0.412·11-s + 0.871·12-s + 1.33·13-s − 0.386·14-s + 0.571·15-s + 0.250·16-s + 1.48·17-s − 1.44·18-s + 1.37·19-s + 0.164·20-s + 0.952·21-s − 0.291·22-s − 0.616·24-s − 0.892·25-s − 0.941·26-s + 1.80·27-s + 0.273·28-s + 0.0229·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.063860126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.063860126\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 9.05T + 27T^{2} \) |
| 5 | \( 1 - 3.66T + 125T^{2} \) |
| 7 | \( 1 - 10.1T + 343T^{2} \) |
| 11 | \( 1 - 15.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 3.57T + 2.43e4T^{2} \) |
| 31 | \( 1 + 323.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 99.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 89.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 435.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 585.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 531.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 50.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 397.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 37.9T + 3.57e5T^{2} \) |
| 73 | \( 1 - 425.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 253.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 702.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 472.T + 9.12e5T^{2} \) |
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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
−9.447876009080429199233311600059, −8.729840297277168152569587471183, −7.87614724686425049100820357718, −7.60936097129348461492322422111, −6.37458882227983320071333499701, −5.24030270955390735665015404196, −3.66441132543563679050531356337, −3.25441223258349461243304361659, −1.83534743982471992701057633670, −1.27899592701710879522336015560,
1.27899592701710879522336015560, 1.83534743982471992701057633670, 3.25441223258349461243304361659, 3.66441132543563679050531356337, 5.24030270955390735665015404196, 6.37458882227983320071333499701, 7.60936097129348461492322422111, 7.87614724686425049100820357718, 8.729840297277168152569587471183, 9.447876009080429199233311600059
