| L(s) = 1 | + 3-s + 3.17·5-s + 7-s + 9-s − 5.06·11-s + 4.07·13-s + 3.17·15-s + 4.22·17-s + 5.06·19-s + 21-s − 23-s + 5.07·25-s + 27-s + 6.68·29-s + 2.22·31-s − 5.06·33-s + 3.17·35-s − 1.91·37-s + 4.07·39-s − 1.37·41-s + 3.39·43-s + 3.17·45-s + 5.81·47-s + 49-s + 4.22·51-s − 6.57·53-s − 16.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.41·5-s + 0.377·7-s + 0.333·9-s − 1.52·11-s + 1.13·13-s + 0.819·15-s + 1.02·17-s + 1.16·19-s + 0.218·21-s − 0.208·23-s + 1.01·25-s + 0.192·27-s + 1.24·29-s + 0.398·31-s − 0.881·33-s + 0.536·35-s − 0.314·37-s + 0.652·39-s − 0.214·41-s + 0.517·43-s + 0.473·45-s + 0.848·47-s + 0.142·49-s + 0.591·51-s − 0.903·53-s − 2.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.992243708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.992243708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.17T + 5T^{2} \) |
| 11 | \( 1 + 5.06T + 11T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 17 | \( 1 - 4.22T + 17T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 2.22T + 31T^{2} \) |
| 37 | \( 1 + 1.91T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 - 3.39T + 43T^{2} \) |
| 47 | \( 1 - 5.81T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 - 5.67T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 2.95T + 71T^{2} \) |
| 73 | \( 1 + 2.02T + 73T^{2} \) |
| 79 | \( 1 + 7.14T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 2.08T + 97T^{2} \) |
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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
−7.926813277136337257696804711769, −7.33050708722820324285181231296, −6.33507335886627301921727926713, −5.69567854999231425098470857562, −5.24519347285259693068286983220, −4.39317645585282890941837901331, −3.17974809303916313369342021820, −2.77608636978176645962163344564, −1.78647150633728923122749495521, −1.04912096756672502406369321281,
1.04912096756672502406369321281, 1.78647150633728923122749495521, 2.77608636978176645962163344564, 3.17974809303916313369342021820, 4.39317645585282890941837901331, 5.24519347285259693068286983220, 5.69567854999231425098470857562, 6.33507335886627301921727926713, 7.33050708722820324285181231296, 7.926813277136337257696804711769
