| L(s) = 1 | + 2-s + 4-s − 1.41·5-s − 4.24·7-s + 8-s − 1.41·10-s − 2.82·11-s + 2·13-s − 4.24·14-s + 16-s − 2·19-s − 1.41·20-s − 2.82·22-s − 7.07·23-s − 2.99·25-s + 2·26-s − 4.24·28-s + 7.07·29-s − 4.24·31-s + 32-s + 6·35-s + 4.24·37-s − 2·38-s − 1.41·40-s + 5.65·41-s + 4·43-s − 2.82·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.632·5-s − 1.60·7-s + 0.353·8-s − 0.447·10-s − 0.852·11-s + 0.554·13-s − 1.13·14-s + 0.250·16-s − 0.458·19-s − 0.316·20-s − 0.603·22-s − 1.47·23-s − 0.599·25-s + 0.392·26-s − 0.801·28-s + 1.31·29-s − 0.762·31-s + 0.176·32-s + 1.01·35-s + 0.697·37-s − 0.324·38-s − 0.223·40-s + 0.883·41-s + 0.609·43-s − 0.426·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.606325182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.606325182\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8.48T + 97T^{2} \) |
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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
−8.049655843218431989021327245434, −7.41513880903610447701056551660, −6.63733405985865564790688664745, −5.98842677906479590239643103831, −5.48102539109152360512203043060, −4.12203555595683966032328409253, −3.95526978722396705200356380772, −2.93146050657486579241672829640, −2.28884633619487860030961178471, −0.58247694876350122164203597616,
0.58247694876350122164203597616, 2.28884633619487860030961178471, 2.93146050657486579241672829640, 3.95526978722396705200356380772, 4.12203555595683966032328409253, 5.48102539109152360512203043060, 5.98842677906479590239643103831, 6.63733405985865564790688664745, 7.41513880903610447701056551660, 8.049655843218431989021327245434
