| L(s) = 1 | − 2.65·2-s + 1.37·3-s + 5.02·4-s − 3.65·6-s − 2.65·7-s − 8.02·8-s − 1.10·9-s + 0.273·11-s + 6.92·12-s + 0.547·13-s + 7.02·14-s + 11.2·16-s − 2·17-s + 2.92·18-s + 7.84·19-s − 3.65·21-s − 0.726·22-s + 2.20·23-s − 11.0·24-s − 1.45·26-s − 5.65·27-s − 13.3·28-s − 7.84·29-s − 8.15·31-s − 13.7·32-s + 0.377·33-s + 5.30·34-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.795·3-s + 2.51·4-s − 1.49·6-s − 1.00·7-s − 2.83·8-s − 0.367·9-s + 0.0825·11-s + 1.99·12-s + 0.151·13-s + 1.87·14-s + 2.80·16-s − 0.485·17-s + 0.689·18-s + 1.80·19-s − 0.796·21-s − 0.154·22-s + 0.460·23-s − 2.25·24-s − 0.284·26-s − 1.08·27-s − 2.51·28-s − 1.45·29-s − 1.46·31-s − 2.42·32-s + 0.0656·33-s + 0.909·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 - 1.37T + 3T^{2} \) |
| 7 | \( 1 + 2.65T + 7T^{2} \) |
| 11 | \( 1 - 0.273T + 11T^{2} \) |
| 13 | \( 1 - 0.547T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 7.84T + 19T^{2} \) |
| 23 | \( 1 - 2.20T + 23T^{2} \) |
| 29 | \( 1 + 7.84T + 29T^{2} \) |
| 31 | \( 1 + 8.15T + 31T^{2} \) |
| 37 | \( 1 - 1.02T + 37T^{2} \) |
| 41 | \( 1 - 7.58T + 41T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 7.30T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 4.54T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 + 2.75T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 1.11T + 83T^{2} \) |
| 89 | \( 1 + 9.15T + 89T^{2} \) |
| 97 | \( 1 + 0.206T + 97T^{2} \) |
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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
−9.480432917552389253442251221845, −8.905607167516549617096817182881, −7.899067871030726676839105298391, −7.40249744872997720370641826249, −6.49371453202045227753720103928, −5.56860306345129310180024216912, −3.47651934574139663742155345870, −2.83819872798380508816397573618, −1.61995318054914676619695637115, 0,
1.61995318054914676619695637115, 2.83819872798380508816397573618, 3.47651934574139663742155345870, 5.56860306345129310180024216912, 6.49371453202045227753720103928, 7.40249744872997720370641826249, 7.899067871030726676839105298391, 8.905607167516549617096817182881, 9.480432917552389253442251221845
