| L(s) = 1 | − 0.311·2-s + 1.90·3-s − 1.90·4-s − 2.31·5-s − 0.592·6-s + 2.90·7-s + 1.21·8-s + 0.622·9-s + 0.719·10-s − 5.52·11-s − 3.62·12-s + 13-s − 0.903·14-s − 4.39·15-s + 3.42·16-s + 4.80·17-s − 0.193·18-s − 3.83·19-s + 4.39·20-s + 5.52·21-s + 1.71·22-s + 5.67·23-s + 2.31·24-s + 0.341·25-s − 0.311·26-s − 4.52·27-s − 5.52·28-s + ⋯ |
| L(s) = 1 | − 0.219·2-s + 1.09·3-s − 0.951·4-s − 1.03·5-s − 0.241·6-s + 1.09·7-s + 0.429·8-s + 0.207·9-s + 0.227·10-s − 1.66·11-s − 1.04·12-s + 0.277·13-s − 0.241·14-s − 1.13·15-s + 0.857·16-s + 1.16·17-s − 0.0456·18-s − 0.880·19-s + 0.983·20-s + 1.20·21-s + 0.366·22-s + 1.18·23-s + 0.471·24-s + 0.0682·25-s − 0.0610·26-s − 0.870·27-s − 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 3 | \( 1 - 1.90T + 3T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 7 | \( 1 - 2.90T + 7T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 - 5.67T + 23T^{2} \) |
| 29 | \( 1 + 7.90T + 29T^{2} \) |
| 31 | \( 1 + 7.18T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 + 7.05T + 41T^{2} \) |
| 43 | \( 1 - 2.62T + 43T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 + 6.28T + 53T^{2} \) |
| 59 | \( 1 + 7.98T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 - 2.75T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 1.86T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 2.94T + 97T^{2} \) |
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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
−8.944453343451633870504240753696, −8.287515655668730613103611923475, −7.86234610952019325963245892582, −7.42134567138561265504589502246, −5.49073007500430062813703324406, −4.93337076943410095699950113503, −3.84616131678550058788098321201, −3.17737415089227285402559236251, −1.78878245489352560395458767859, 0,
1.78878245489352560395458767859, 3.17737415089227285402559236251, 3.84616131678550058788098321201, 4.93337076943410095699950113503, 5.49073007500430062813703324406, 7.42134567138561265504589502246, 7.86234610952019325963245892582, 8.287515655668730613103611923475, 8.944453343451633870504240753696
