| L(s) = 1 | + 2-s + 0.888·3-s + 4-s − 3.21·5-s + 0.888·6-s + 3.29·7-s + 8-s − 2.21·9-s − 3.21·10-s + 5.53·11-s + 0.888·12-s − 2.65·13-s + 3.29·14-s − 2.85·15-s + 16-s − 4.52·17-s − 2.21·18-s + 7.97·19-s − 3.21·20-s + 2.93·21-s + 5.53·22-s + 0.739·23-s + 0.888·24-s + 5.35·25-s − 2.65·26-s − 4.62·27-s + 3.29·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.513·3-s + 0.5·4-s − 1.43·5-s + 0.362·6-s + 1.24·7-s + 0.353·8-s − 0.736·9-s − 1.01·10-s + 1.66·11-s + 0.256·12-s − 0.737·13-s + 0.881·14-s − 0.738·15-s + 0.250·16-s − 1.09·17-s − 0.521·18-s + 1.83·19-s − 0.719·20-s + 0.639·21-s + 1.18·22-s + 0.154·23-s + 0.181·24-s + 1.07·25-s − 0.521·26-s − 0.890·27-s + 0.623·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.317913861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.317913861\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 0.888T + 3T^{2} \) |
| 5 | \( 1 + 3.21T + 5T^{2} \) |
| 7 | \( 1 - 3.29T + 7T^{2} \) |
| 11 | \( 1 - 5.53T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 - 7.97T + 19T^{2} \) |
| 23 | \( 1 - 0.739T + 23T^{2} \) |
| 29 | \( 1 + 5.39T + 29T^{2} \) |
| 31 | \( 1 - 6.00T + 31T^{2} \) |
| 37 | \( 1 - 9.75T + 37T^{2} \) |
| 41 | \( 1 + 0.718T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 - 6.20T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 6.78T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 8.03T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 0.859T + 89T^{2} \) |
| 97 | \( 1 + 19.0T + 97T^{2} \) |
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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
−8.231144848879363056034476539160, −7.70646556238425603272912674726, −7.16662188344422981013595401118, −6.24326199609388251293012019838, −5.21533175292270809534904196736, −4.49885070159045476557972126331, −3.93092191767188947406190425763, −3.16980488384833467001934754277, −2.18596637880483004868106984235, −0.949142615096396442199366341007,
0.949142615096396442199366341007, 2.18596637880483004868106984235, 3.16980488384833467001934754277, 3.93092191767188947406190425763, 4.49885070159045476557972126331, 5.21533175292270809534904196736, 6.24326199609388251293012019838, 7.16662188344422981013595401118, 7.70646556238425603272912674726, 8.231144848879363056034476539160
