| L(s) = 1 | + 2-s + 2.86·3-s + 4-s − 3.32·5-s + 2.86·6-s + 2.39·7-s + 8-s + 5.18·9-s − 3.32·10-s + 2.86·12-s + 5.32·13-s + 2.39·14-s − 9.50·15-s + 16-s − 2.92·17-s + 5.18·18-s + 1.13·19-s − 3.32·20-s + 6.86·21-s + 4.18·23-s + 2.86·24-s + 6.04·25-s + 5.32·26-s + 6.24·27-s + 2.39·28-s + 8.98·29-s − 9.50·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.65·3-s + 0.5·4-s − 1.48·5-s + 1.16·6-s + 0.906·7-s + 0.353·8-s + 1.72·9-s − 1.05·10-s + 0.825·12-s + 1.47·13-s + 0.640·14-s − 2.45·15-s + 0.250·16-s − 0.709·17-s + 1.22·18-s + 0.261·19-s − 0.743·20-s + 1.49·21-s + 0.872·23-s + 0.583·24-s + 1.20·25-s + 1.04·26-s + 1.20·27-s + 0.453·28-s + 1.66·29-s − 1.73·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.176891050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.176891050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 2.86T + 3T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 13 | \( 1 - 5.32T + 13T^{2} \) |
| 17 | \( 1 + 2.92T + 17T^{2} \) |
| 19 | \( 1 - 1.13T + 19T^{2} \) |
| 23 | \( 1 - 4.18T + 23T^{2} \) |
| 29 | \( 1 - 8.98T + 29T^{2} \) |
| 31 | \( 1 + 7.97T + 31T^{2} \) |
| 37 | \( 1 + 8.46T + 37T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 - 7.32T + 47T^{2} \) |
| 53 | \( 1 - 4.18T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.98T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 - 9.25T + 71T^{2} \) |
| 73 | \( 1 + 7.69T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 1.19T + 83T^{2} \) |
| 89 | \( 1 - 4.83T + 89T^{2} \) |
| 97 | \( 1 - 7.69T + 97T^{2} \) |
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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
−7.56054707452762492222895148832, −7.34682632040868961990705494455, −6.47888378471403687037732849500, −5.41393857088806988420274233739, −4.44780521870867008619723791541, −4.14379795868847741406991390191, −3.43027284669914255138095726763, −2.92434926650851322019369569189, −1.93940135877628079561213969274, −1.04137524719411801388393300660,
1.04137524719411801388393300660, 1.93940135877628079561213969274, 2.92434926650851322019369569189, 3.43027284669914255138095726763, 4.14379795868847741406991390191, 4.44780521870867008619723791541, 5.41393857088806988420274233739, 6.47888378471403687037732849500, 7.34682632040868961990705494455, 7.56054707452762492222895148832
