| L(s) = 1 | − 2.49·2-s + 3-s + 4.21·4-s − 2.49·6-s − 1.92·7-s − 5.52·8-s + 9-s + 4.21·12-s + 2.28·13-s + 4.78·14-s + 5.33·16-s + 3.60·17-s − 2.49·18-s − 0.936·19-s − 1.92·21-s + 9.34·23-s − 5.52·24-s − 5.70·26-s + 27-s − 8.09·28-s − 6.91·29-s − 4.59·31-s − 2.26·32-s − 8.98·34-s + 4.21·36-s + 8.31·37-s + 2.33·38-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.577·3-s + 2.10·4-s − 1.01·6-s − 0.725·7-s − 1.95·8-s + 0.333·9-s + 1.21·12-s + 0.634·13-s + 1.27·14-s + 1.33·16-s + 0.873·17-s − 0.587·18-s − 0.214·19-s − 0.419·21-s + 1.94·23-s − 1.12·24-s − 1.11·26-s + 0.192·27-s − 1.53·28-s − 1.28·29-s − 0.824·31-s − 0.400·32-s − 1.54·34-s + 0.702·36-s + 1.36·37-s + 0.378·38-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 | \( 1 + 2.49T + 2T^{2} \) |
| 7 | \( 1 + 1.92T + 7T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 + 0.936T + 19T^{2} \) |
| 23 | \( 1 - 9.34T + 23T^{2} \) |
| 29 | \( 1 + 6.91T + 29T^{2} \) |
| 31 | \( 1 + 4.59T + 31T^{2} \) |
| 37 | \( 1 - 8.31T + 37T^{2} \) |
| 41 | \( 1 + 7.56T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 - 1.97T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 3.38T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 0.231T + 67T^{2} \) |
| 71 | \( 1 + 3.45T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 2.50T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 8.89T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55389137725996284589121636398, −7.04054344038278105117797973177, −6.35751553692287794828193883664, −5.64459629474415775343655657737, −4.51910809443510702025058390678, −3.25851113480248710240547826297, −3.05556398211836298507384934389, −1.83305899874286802448437321953, −1.17829131338712605919341446540, 0,
1.17829131338712605919341446540, 1.83305899874286802448437321953, 3.05556398211836298507384934389, 3.25851113480248710240547826297, 4.51910809443510702025058390678, 5.64459629474415775343655657737, 6.35751553692287794828193883664, 7.04054344038278105117797973177, 7.55389137725996284589121636398
