| L(s) = 1 | − 0.311·2-s − 1.90·4-s − 0.903·7-s + 1.21·8-s − 11-s + 2.90·13-s + 0.280·14-s + 3.42·16-s − 2.28·17-s + 2.42·19-s + 0.311·22-s − 4·23-s − 0.903·26-s + 1.71·28-s − 7.05·29-s − 2.62·31-s − 3.49·32-s + 0.709·34-s + 5.80·37-s − 0.755·38-s + 10.6·41-s + 10.7·43-s + 1.90·44-s + 1.24·46-s − 0.949·47-s − 6.18·49-s − 5.52·52-s + ⋯ |
| L(s) = 1 | − 0.219·2-s − 0.951·4-s − 0.341·7-s + 0.429·8-s − 0.301·11-s + 0.805·13-s + 0.0750·14-s + 0.857·16-s − 0.553·17-s + 0.557·19-s + 0.0663·22-s − 0.834·23-s − 0.177·26-s + 0.324·28-s − 1.30·29-s − 0.470·31-s − 0.617·32-s + 0.121·34-s + 0.954·37-s − 0.122·38-s + 1.66·41-s + 1.63·43-s + 0.286·44-s + 0.183·46-s − 0.138·47-s − 0.883·49-s − 0.766·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 7 | \( 1 + 0.903T + 7T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.05T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 0.949T + 47T^{2} \) |
| 53 | \( 1 + 0.815T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 + 7.24T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 9.28T + 71T^{2} \) |
| 73 | \( 1 + 5.65T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 7.76T + 83T^{2} \) |
| 89 | \( 1 + 6.13T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.596272307732239440995307311659, −7.88801868876532845347201275928, −7.20544763931564550459593628254, −5.99480260065212538201931419066, −5.55926408709218094576889273243, −4.36525600916256382587200883267, −3.86216849940829203358962371447, −2.72533408371881332165623795378, −1.34183701200788561466535288150, 0,
1.34183701200788561466535288150, 2.72533408371881332165623795378, 3.86216849940829203358962371447, 4.36525600916256382587200883267, 5.55926408709218094576889273243, 5.99480260065212538201931419066, 7.20544763931564550459593628254, 7.88801868876532845347201275928, 8.596272307732239440995307311659
