| L(s) = 1 | − 1.38·2-s − 3-s − 0.0822·4-s − 0.467·5-s + 1.38·6-s − 3.78·7-s + 2.88·8-s + 9-s + 0.646·10-s + 1.56·11-s + 0.0822·12-s + 13-s + 5.23·14-s + 0.467·15-s − 3.82·16-s + 3.54·17-s − 1.38·18-s − 7.39·19-s + 0.0384·20-s + 3.78·21-s − 2.16·22-s + 1.72·23-s − 2.88·24-s − 4.78·25-s − 1.38·26-s − 27-s + 0.311·28-s + ⋯ |
| L(s) = 1 | − 0.979·2-s − 0.577·3-s − 0.0411·4-s − 0.208·5-s + 0.565·6-s − 1.43·7-s + 1.01·8-s + 0.333·9-s + 0.204·10-s + 0.471·11-s + 0.0237·12-s + 0.277·13-s + 1.40·14-s + 0.120·15-s − 0.957·16-s + 0.860·17-s − 0.326·18-s − 1.69·19-s + 0.00859·20-s + 0.825·21-s − 0.462·22-s + 0.360·23-s − 0.588·24-s − 0.956·25-s − 0.271·26-s − 0.192·27-s + 0.0588·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.38T + 2T^{2} \) |
| 5 | \( 1 + 0.467T + 5T^{2} \) |
| 7 | \( 1 + 3.78T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 + 7.39T + 19T^{2} \) |
| 23 | \( 1 - 1.72T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 - 2.89T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 + 7.64T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 4.19T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 9.49T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 9.44T + 71T^{2} \) |
| 73 | \( 1 - 7.51T + 73T^{2} \) |
| 79 | \( 1 - 0.201T + 79T^{2} \) |
| 83 | \( 1 - 0.181T + 83T^{2} \) |
| 89 | \( 1 + 4.28T + 89T^{2} \) |
| 97 | \( 1 - 5.99T + 97T^{2} \) |
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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
−8.331846990876430050227171727028, −7.29957714724874140927830400851, −6.71641477614287688449680439839, −6.08771096583469258352019552171, −5.16802730418719574855651572313, −4.11157802156230834642322795383, −3.57715791703964815259110502804, −2.22272438507117571205286387629, −0.959500234937909861060775036605, 0,
0.959500234937909861060775036605, 2.22272438507117571205286387629, 3.57715791703964815259110502804, 4.11157802156230834642322795383, 5.16802730418719574855651572313, 6.08771096583469258352019552171, 6.71641477614287688449680439839, 7.29957714724874140927830400851, 8.331846990876430050227171727028
