| L(s) = 1 | − 4·7-s − 72·11-s + 6·13-s + 38·17-s + 52·19-s + 152·23-s + 78·29-s + 120·31-s + 150·37-s − 362·41-s + 484·43-s + 280·47-s − 327·49-s − 670·53-s − 696·59-s + 222·61-s + 4·67-s − 96·71-s − 178·73-s + 288·77-s − 632·79-s − 612·83-s − 994·89-s − 24·91-s − 1.63e3·97-s − 890·101-s + 524·103-s + ⋯ |
| L(s) = 1 | − 0.215·7-s − 1.97·11-s + 0.128·13-s + 0.542·17-s + 0.627·19-s + 1.37·23-s + 0.499·29-s + 0.695·31-s + 0.666·37-s − 1.37·41-s + 1.71·43-s + 0.868·47-s − 0.953·49-s − 1.73·53-s − 1.53·59-s + 0.465·61-s + 0.00729·67-s − 0.160·71-s − 0.285·73-s + 0.426·77-s − 0.900·79-s − 0.809·83-s − 1.18·89-s − 0.0276·91-s − 1.71·97-s − 0.876·101-s + 0.501·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 72 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 120 T + p^{3} T^{2} \) |
| 37 | \( 1 - 150 T + p^{3} T^{2} \) |
| 41 | \( 1 + 362 T + p^{3} T^{2} \) |
| 43 | \( 1 - 484 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 + 670 T + p^{3} T^{2} \) |
| 59 | \( 1 + 696 T + p^{3} T^{2} \) |
| 61 | \( 1 - 222 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 T + p^{3} T^{2} \) |
| 71 | \( 1 + 96 T + p^{3} T^{2} \) |
| 73 | \( 1 + 178 T + p^{3} T^{2} \) |
| 79 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 + 994 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1634 T + p^{3} T^{2} \) |
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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
−8.390971251333930408097571222007, −7.76332397269337943536682979597, −7.07236608091361917389327448153, −6.00062944942253497167779779432, −5.24912315752427130523143466724, −4.55310878382829645275085282976, −3.14485613788732602838121053056, −2.67881017750003495627325756295, −1.20308913548803146509199498068, 0,
1.20308913548803146509199498068, 2.67881017750003495627325756295, 3.14485613788732602838121053056, 4.55310878382829645275085282976, 5.24912315752427130523143466724, 6.00062944942253497167779779432, 7.07236608091361917389327448153, 7.76332397269337943536682979597, 8.390971251333930408097571222007
