| L(s) = 1 | − 1.03·2-s − 3-s − 0.924·4-s − 1.10·5-s + 1.03·6-s − 7-s + 3.03·8-s + 9-s + 1.14·10-s − 3.63·11-s + 0.924·12-s − 3.11·13-s + 1.03·14-s + 1.10·15-s − 1.29·16-s − 0.947·17-s − 1.03·18-s − 5.88·19-s + 1.02·20-s + 21-s + 3.76·22-s − 4.48·23-s − 3.03·24-s − 3.77·25-s + 3.23·26-s − 27-s + 0.924·28-s + ⋯ |
| L(s) = 1 | − 0.733·2-s − 0.577·3-s − 0.462·4-s − 0.494·5-s + 0.423·6-s − 0.377·7-s + 1.07·8-s + 0.333·9-s + 0.362·10-s − 1.09·11-s + 0.266·12-s − 0.864·13-s + 0.277·14-s + 0.285·15-s − 0.324·16-s − 0.229·17-s − 0.244·18-s − 1.34·19-s + 0.228·20-s + 0.218·21-s + 0.803·22-s − 0.934·23-s − 0.619·24-s − 0.755·25-s + 0.633·26-s − 0.192·27-s + 0.174·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05294581646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05294581646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 1.03T + 2T^{2} \) |
| 5 | \( 1 + 1.10T + 5T^{2} \) |
| 11 | \( 1 + 3.63T + 11T^{2} \) |
| 13 | \( 1 + 3.11T + 13T^{2} \) |
| 17 | \( 1 + 0.947T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 + 4.48T + 23T^{2} \) |
| 29 | \( 1 + 0.208T + 29T^{2} \) |
| 31 | \( 1 + 2.92T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 4.40T + 41T^{2} \) |
| 43 | \( 1 + 5.58T + 43T^{2} \) |
| 47 | \( 1 + 6.89T + 47T^{2} \) |
| 53 | \( 1 - 8.55T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 2.33T + 73T^{2} \) |
| 79 | \( 1 + 3.93T + 79T^{2} \) |
| 83 | \( 1 + 8.71T + 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 - 1.19T + 97T^{2} \) |
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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
−8.368831681378593807354135933885, −7.75652455441092031649984640699, −7.28783986842457777761637077364, −6.22465083912349730372706109875, −5.52404681710221673223362271023, −4.49190161681851934106839605506, −4.19910999572125560187355359936, −2.82660061570134255269478601659, −1.75252855956299290700112130566, −0.14734895379000028106471661988,
0.14734895379000028106471661988, 1.75252855956299290700112130566, 2.82660061570134255269478601659, 4.19910999572125560187355359936, 4.49190161681851934106839605506, 5.52404681710221673223362271023, 6.22465083912349730372706109875, 7.28783986842457777761637077364, 7.75652455441092031649984640699, 8.368831681378593807354135933885
