| L(s) = 1 | + 0.639·5-s − 3·9-s + 5.62i·11-s − 6.89i·19-s + 4.79i·23-s − 4.59·25-s + 9.59i·31-s − 11.8·37-s − 9.59·41-s − 4.34i·43-s − 1.91·45-s + 2i·47-s − 7·49-s + 13.1·53-s + 3.59i·55-s + ⋯ |
| L(s) = 1 | + 0.285·5-s − 9-s + 1.69i·11-s − 1.58i·19-s + 0.999i·23-s − 0.918·25-s + 1.72i·31-s − 1.95·37-s − 1.49·41-s − 0.662i·43-s − 0.285·45-s + 0.291i·47-s − 49-s + 1.80·53-s + 0.484i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7696377878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7696377878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - 4.79iT \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 0.639T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 5.62iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6.89iT - 19T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 9.59iT - 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 + 9.59T + 41T^{2} \) |
| 43 | \( 1 + 4.34iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 8.17iT - 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 + 9.59T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 18.1iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.830186301173117474698481302097, −8.977189571030249200849415878206, −8.369019029327540087148167443419, −7.08636825892124737737664548666, −6.87657549539303278364056108937, −5.43598114201936707694232313611, −5.03609117129367939065371536251, −3.80727114607320931391896797420, −2.68949170469390789328204407063, −1.69949810424729057754076709054,
0.28558121446666688944813274362, 1.93704521459027060719711839930, 3.14800840907423645696040757902, 3.86435049453839051409941504765, 5.30551827694621050131214570488, 5.89363563162654211348192069677, 6.50687356795270449981762326170, 7.88567382679192589733783821096, 8.383647894328181785666872514076, 9.045184971500424898293847846066
