| L(s) = 1 | + 3-s − 0.649·7-s + 9-s + 5.35·11-s + 0.343·13-s + 5.13·17-s − 1.53·19-s − 0.649·21-s + 6.81·23-s + 27-s + 1.68·29-s − 31-s + 5.35·33-s − 4.23·37-s + 0.343·39-s + 4.30·41-s − 4.66·43-s − 3.25·47-s − 6.57·49-s + 5.13·51-s + 5.24·53-s − 1.53·57-s + 13.9·59-s + 4.53·61-s − 0.649·63-s + 0.342·67-s + 6.81·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.245·7-s + 0.333·9-s + 1.61·11-s + 0.0951·13-s + 1.24·17-s − 0.351·19-s − 0.141·21-s + 1.42·23-s + 0.192·27-s + 0.312·29-s − 0.179·31-s + 0.932·33-s − 0.696·37-s + 0.0549·39-s + 0.672·41-s − 0.710·43-s − 0.475·47-s − 0.939·49-s + 0.718·51-s + 0.720·53-s − 0.202·57-s + 1.81·59-s + 0.581·61-s − 0.0818·63-s + 0.0418·67-s + 0.820·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.246849863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.246849863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 0.649T + 7T^{2} \) |
| 11 | \( 1 - 5.35T + 11T^{2} \) |
| 13 | \( 1 - 0.343T + 13T^{2} \) |
| 17 | \( 1 - 5.13T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 - 4.30T + 41T^{2} \) |
| 43 | \( 1 + 4.66T + 43T^{2} \) |
| 47 | \( 1 + 3.25T + 47T^{2} \) |
| 53 | \( 1 - 5.24T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 4.53T + 61T^{2} \) |
| 67 | \( 1 - 0.342T + 67T^{2} \) |
| 71 | \( 1 - 3.50T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 1.64T + 79T^{2} \) |
| 83 | \( 1 + 8.10T + 83T^{2} \) |
| 89 | \( 1 + 8.61T + 89T^{2} \) |
| 97 | \( 1 + 7.92T + 97T^{2} \) |
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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
−7.71946036407636091167542494535, −6.89356102333151014435478883343, −6.61515258454347277357809819155, −5.65300254847519915257922155514, −4.92603856642132129021571676463, −3.97866531722589315941868037365, −3.50963162257486316841622062279, −2.74305782640869013885560474945, −1.64089886615134634070263618455, −0.912214646054641848865846232290,
0.912214646054641848865846232290, 1.64089886615134634070263618455, 2.74305782640869013885560474945, 3.50963162257486316841622062279, 3.97866531722589315941868037365, 4.92603856642132129021571676463, 5.65300254847519915257922155514, 6.61515258454347277357809819155, 6.89356102333151014435478883343, 7.71946036407636091167542494535
