| L(s) = 1 | + 2.79·3-s + 5-s − 1.79·7-s + 4.79·9-s + 0.791·11-s − 5.79·13-s + 2.79·15-s + 0.791·17-s − 5.79·19-s − 5·21-s + 23-s + 25-s + 4.99·27-s − 7.58·29-s − 3.37·31-s + 2.20·33-s − 1.79·35-s + 4·37-s − 16.1·39-s − 6.79·41-s − 11.1·43-s + 4.79·45-s − 4.41·47-s − 3.79·49-s + 2.20·51-s − 6·53-s + 0.791·55-s + ⋯ |
| L(s) = 1 | + 1.61·3-s + 0.447·5-s − 0.677·7-s + 1.59·9-s + 0.238·11-s − 1.60·13-s + 0.720·15-s + 0.191·17-s − 1.32·19-s − 1.09·21-s + 0.208·23-s + 0.200·25-s + 0.962·27-s − 1.40·29-s − 0.605·31-s + 0.384·33-s − 0.302·35-s + 0.657·37-s − 2.58·39-s − 1.06·41-s − 1.70·43-s + 0.714·45-s − 0.644·47-s − 0.541·49-s + 0.309·51-s − 0.824·53-s + 0.106·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 7 | \( 1 + 1.79T + 7T^{2} \) |
| 11 | \( 1 - 0.791T + 11T^{2} \) |
| 13 | \( 1 + 5.79T + 13T^{2} \) |
| 17 | \( 1 - 0.791T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 29 | \( 1 + 7.58T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 7.95T + 97T^{2} \) |
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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
−7.72559270071860145983813959721, −6.90538024139718686630943959359, −6.45618258307858548893260869478, −5.33093501242736514766254819710, −4.59617821246778085282063405507, −3.66439985493917045119517114312, −3.13068602656314140045396435743, −2.24434515295162159333032384585, −1.79253750151979273035453353504, 0,
1.79253750151979273035453353504, 2.24434515295162159333032384585, 3.13068602656314140045396435743, 3.66439985493917045119517114312, 4.59617821246778085282063405507, 5.33093501242736514766254819710, 6.45618258307858548893260869478, 6.90538024139718686630943959359, 7.72559270071860145983813959721
