| L(s) = 1 | − 2.66·3-s − 0.750·7-s + 4.09·9-s + 4.57·11-s + 3.84·13-s + 7.32·17-s − 6.57·19-s + 2.00·21-s + 23-s − 2.91·27-s − 7.00·29-s − 4.43·31-s − 12.1·33-s − 0.860·37-s − 10.2·39-s − 9.60·41-s + 6.57·43-s − 10.9·47-s − 6.43·49-s − 19.5·51-s + 1.82·53-s + 17.5·57-s − 7.57·59-s + 9.82·61-s − 3.07·63-s − 10.1·67-s − 2.66·69-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 0.283·7-s + 1.36·9-s + 1.37·11-s + 1.06·13-s + 1.77·17-s − 1.50·19-s + 0.436·21-s + 0.208·23-s − 0.560·27-s − 1.30·29-s − 0.795·31-s − 2.12·33-s − 0.141·37-s − 1.63·39-s − 1.49·41-s + 1.00·43-s − 1.59·47-s − 0.919·49-s − 2.73·51-s + 0.250·53-s + 2.31·57-s − 0.986·59-s + 1.25·61-s − 0.387·63-s − 1.24·67-s − 0.320·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.66T + 3T^{2} \) |
| 7 | \( 1 + 0.750T + 7T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 - 3.84T + 13T^{2} \) |
| 17 | \( 1 - 7.32T + 17T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 + 0.860T + 37T^{2} \) |
| 41 | \( 1 + 9.60T + 41T^{2} \) |
| 43 | \( 1 - 6.57T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 1.82T + 53T^{2} \) |
| 59 | \( 1 + 7.57T + 59T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 4.42T + 71T^{2} \) |
| 73 | \( 1 - 4.50T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 3.86T + 89T^{2} \) |
| 97 | \( 1 - 0.537T + 97T^{2} \) |
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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
−7.10829170108391086943350890685, −6.46299330832273374076475841172, −6.08693460683228492964696577901, −5.48259988318640340319384096536, −4.72593808753428421227435772288, −3.81763294158323550175992442635, −3.41018308033383126866490991289, −1.77078122827618100345428196725, −1.13631317048332933303184451211, 0,
1.13631317048332933303184451211, 1.77078122827618100345428196725, 3.41018308033383126866490991289, 3.81763294158323550175992442635, 4.72593808753428421227435772288, 5.48259988318640340319384096536, 6.08693460683228492964696577901, 6.46299330832273374076475841172, 7.10829170108391086943350890685
