| L(s) = 1 | − 2.72·2-s + 3-s + 5.43·4-s − 2.72·6-s − 0.486·7-s − 9.35·8-s + 9-s + 5.43·12-s − 3.61·13-s + 1.32·14-s + 14.6·16-s + 3.64·17-s − 2.72·18-s − 1.63·19-s − 0.486·21-s + 3.54·23-s − 9.35·24-s + 9.84·26-s + 27-s − 2.64·28-s + 1.05·29-s + 6.10·31-s − 21.1·32-s − 9.93·34-s + 5.43·36-s − 8.59·37-s + 4.44·38-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 0.577·3-s + 2.71·4-s − 1.11·6-s − 0.183·7-s − 3.30·8-s + 0.333·9-s + 1.56·12-s − 1.00·13-s + 0.354·14-s + 3.65·16-s + 0.883·17-s − 0.642·18-s − 0.374·19-s − 0.106·21-s + 0.738·23-s − 1.90·24-s + 1.93·26-s + 0.192·27-s − 0.499·28-s + 0.196·29-s + 1.09·31-s − 3.74·32-s − 1.70·34-s + 0.905·36-s − 1.41·37-s + 0.721·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 7 | \( 1 + 0.486T + 7T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 - 3.64T + 17T^{2} \) |
| 19 | \( 1 + 1.63T + 19T^{2} \) |
| 23 | \( 1 - 3.54T + 23T^{2} \) |
| 29 | \( 1 - 1.05T + 29T^{2} \) |
| 31 | \( 1 - 6.10T + 31T^{2} \) |
| 37 | \( 1 + 8.59T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 7.41T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 - 1.49T + 53T^{2} \) |
| 59 | \( 1 + 7.97T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 0.432T + 67T^{2} \) |
| 71 | \( 1 - 6.01T + 71T^{2} \) |
| 73 | \( 1 - 1.16T + 73T^{2} \) |
| 79 | \( 1 + 4.42T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 - 1.38T + 89T^{2} \) |
| 97 | \( 1 - 1.39T + 97T^{2} \) |
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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
−7.71514775996925202240182911414, −6.93841260300832176650188013151, −6.51687527755180834014686420755, −5.57851867095255474591584374858, −4.60789371654349052117133895231, −3.28312569520886691331813608785, −2.82670679043855055705797481350, −1.94784179144069247587037766179, −1.13442091955988831376598896782, 0,
1.13442091955988831376598896782, 1.94784179144069247587037766179, 2.82670679043855055705797481350, 3.28312569520886691331813608785, 4.60789371654349052117133895231, 5.57851867095255474591584374858, 6.51687527755180834014686420755, 6.93841260300832176650188013151, 7.71514775996925202240182911414
