| L(s) = 1 | − 2·4-s + (1.22 − 1.87i)5-s − 2.82i·11-s + 2.64·13-s + 4·16-s − 3.74i·17-s − 1.73i·19-s + (−2.44 + 3.74i)20-s − 6.48·23-s + (−2 − 4.58i)25-s + 1.41i·29-s + 5.19i·31-s + 4.58i·37-s + 4.89·41-s − 4.58i·43-s + 5.65i·44-s + ⋯ |
| L(s) = 1 | − 4-s + (0.547 − 0.836i)5-s − 0.852i·11-s + 0.733·13-s + 16-s − 0.907i·17-s − 0.397i·19-s + (−0.547 + 0.836i)20-s − 1.35·23-s + (−0.400 − 0.916i)25-s + 0.262i·29-s + 0.933i·31-s + 0.753i·37-s + 0.765·41-s − 0.698i·43-s + 0.852i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.050163750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.050163750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.22 + 1.87i)T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 + 3.74iT - 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 6.48T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 4.58iT - 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 + 4.58iT - 43T^{2} \) |
| 47 | \( 1 + 3.74iT - 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 + 13.7iT - 67T^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 + 14.9iT - 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
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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
−8.679252085611135369472149075698, −8.395387502243485756879006123810, −7.31767435292985437414169052554, −6.10026866652383353921188589585, −5.57981743893404732053745586574, −4.75514170954860371602700545995, −3.99377861310697333037189301413, −2.96128670198283719003690460339, −1.47901301333700946686694006045, −0.39827420464113891480754984352,
1.48028027423828420064776310880, 2.57448179599202634735217712692, 3.88077070114952506038219789246, 4.24069355147805438821952829173, 5.67773903105972549869237684019, 5.95142813486148328786760097237, 7.04307298160157711611613849884, 7.905754421042482199964340024925, 8.542513959589201303388899811108, 9.575460360539384107945054359125
