| L(s) = 1 | + 0.896i·5-s − 4.46i·7-s + 2.96·11-s + 5.46·13-s − 7.20i·17-s − i·19-s − 0.378·23-s + 4.19·25-s − 8.48i·29-s + 7.46i·31-s + 4.00·35-s + 9.46·37-s + 2.82i·41-s + 2.26i·43-s − 8.24·47-s + ⋯ |
| L(s) = 1 | + 0.400i·5-s − 1.68i·7-s + 0.894·11-s + 1.51·13-s − 1.74i·17-s − 0.229i·19-s − 0.0790·23-s + 0.839·25-s − 1.57i·29-s + 1.34i·31-s + 0.676·35-s + 1.55·37-s + 0.441i·41-s + 0.345i·43-s − 1.20·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.272075430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.272075430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.896iT - 5T^{2} \) |
| 7 | \( 1 + 4.46iT - 7T^{2} \) |
| 11 | \( 1 - 2.96T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 + 7.20iT - 17T^{2} \) |
| 23 | \( 1 + 0.378T + 23T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 - 7.46iT - 31T^{2} \) |
| 37 | \( 1 - 9.46T + 37T^{2} \) |
| 41 | \( 1 - 2.82iT - 41T^{2} \) |
| 43 | \( 1 - 2.26iT - 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 + 7.45iT - 53T^{2} \) |
| 59 | \( 1 - 3.10T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 - 4.92iT - 67T^{2} \) |
| 71 | \( 1 + 9.52T + 71T^{2} \) |
| 73 | \( 1 - 7.92T + 73T^{2} \) |
| 79 | \( 1 - 11.4iT - 79T^{2} \) |
| 83 | \( 1 + 8.10T + 83T^{2} \) |
| 89 | \( 1 + 5.93iT - 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.914644234091837645360167450578, −7.16459835958835650912226070396, −6.67505204900689718581189633378, −6.11453132901304983332189004274, −4.88639785529511592730600959441, −4.26521785059978635328904284088, −3.54760936816751887066150563319, −2.82513250788787207599396553104, −1.34950898155935194867390990814, −0.68562284869731598377702873392,
1.25688294926084183973740376178, 1.92890139269900843887235326065, 3.08445801301410375458771341775, 3.83863570883452916377275231103, 4.66146300370382092518786807937, 5.71393196679014872666194443023, 6.04006869127779564177756981809, 6.61110716355223869154224785594, 7.88141621201252493647645091247, 8.465828460856507265797979222385
