| L(s) = 1 | + 4.31i·5-s + 2.20i·7-s − 4.31·11-s + 4.56·13-s + 3.36i·17-s + i·19-s + 5.48·23-s − 13.5·25-s + 9.28i·29-s + 6.56i·31-s − 9.50·35-s + 4.56·37-s + 6.23i·41-s − 8.82i·43-s − 8.74·47-s + ⋯ |
| L(s) = 1 | + 1.92i·5-s + 0.833i·7-s − 1.29·11-s + 1.26·13-s + 0.814i·17-s + 0.229i·19-s + 1.14·23-s − 2.71·25-s + 1.72i·29-s + 1.17i·31-s − 1.60·35-s + 0.750·37-s + 0.973i·41-s − 1.34i·43-s − 1.27·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.518977690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.518977690\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.31iT - 5T^{2} \) |
| 7 | \( 1 - 2.20iT - 7T^{2} \) |
| 11 | \( 1 + 4.31T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 - 3.36iT - 17T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 - 9.28iT - 29T^{2} \) |
| 31 | \( 1 - 6.56iT - 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 - 6.23iT - 41T^{2} \) |
| 43 | \( 1 + 8.82iT - 43T^{2} \) |
| 47 | \( 1 + 8.74T + 47T^{2} \) |
| 53 | \( 1 + 5.21iT - 53T^{2} \) |
| 59 | \( 1 - 0.660T + 59T^{2} \) |
| 61 | \( 1 - 2.30T + 61T^{2} \) |
| 67 | \( 1 - 0.409iT - 67T^{2} \) |
| 71 | \( 1 - 1.23T + 71T^{2} \) |
| 73 | \( 1 - 1.13T + 73T^{2} \) |
| 79 | \( 1 - 15.6iT - 79T^{2} \) |
| 83 | \( 1 + 8.31T + 83T^{2} \) |
| 89 | \( 1 - 7.82iT - 89T^{2} \) |
| 97 | \( 1 - 4.27T + 97T^{2} \) |
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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
−8.406822991522110260117922012678, −7.85890559217867392088956725352, −6.88450942584815201795225987477, −6.58265167972846913543943282901, −5.69472712425965967087602200772, −5.17218819110259681754161057902, −3.78966546450362743042260766920, −3.14596758504201389906520848088, −2.62420429760745238760436305866, −1.60430809907032753592491903557,
0.45486509119131649979757428978, 1.01341260503162859700088856386, 2.19572585673951589531637574884, 3.35888155010010920837458245126, 4.43086371588223758144107737715, 4.64400247217791312323268431694, 5.59680128460960793995452966971, 6.10658362282969000067153494293, 7.35605227417179589477132802848, 7.87610993181709221633592017797
