| L(s) = 1 | − 3.22·5-s + 6.57i·7-s + 13.8i·11-s + 19.0·13-s − 23.0·17-s + 18.8i·19-s + 13.5i·23-s − 14.5·25-s + 0.752·29-s − 46.4i·31-s − 21.1i·35-s + 2.68·37-s + 34.1·41-s + 20.9i·43-s − 15.4i·47-s + ⋯ |
| L(s) = 1 | − 0.645·5-s + 0.938i·7-s + 1.26i·11-s + 1.46·13-s − 1.35·17-s + 0.991i·19-s + 0.591i·23-s − 0.583·25-s + 0.0259·29-s − 1.49i·31-s − 0.605i·35-s + 0.0724·37-s + 0.832·41-s + 0.486i·43-s − 0.327i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7104432957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7104432957\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.22T + 25T^{2} \) |
| 7 | \( 1 - 6.57iT - 49T^{2} \) |
| 11 | \( 1 - 13.8iT - 121T^{2} \) |
| 13 | \( 1 - 19.0T + 169T^{2} \) |
| 17 | \( 1 + 23.0T + 289T^{2} \) |
| 19 | \( 1 - 18.8iT - 361T^{2} \) |
| 23 | \( 1 - 13.5iT - 529T^{2} \) |
| 29 | \( 1 - 0.752T + 841T^{2} \) |
| 31 | \( 1 + 46.4iT - 961T^{2} \) |
| 37 | \( 1 - 2.68T + 1.36e3T^{2} \) |
| 41 | \( 1 - 34.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 15.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 46.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 40.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 105.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 27.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 24.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 95.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 169.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 93.1T + 9.40e3T^{2} \) |
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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
−9.434315132280818942277259699858, −8.712198197271327697838299590992, −8.011554702383474596126783373901, −7.23670304017005975553879135983, −6.22560248530860639672660567209, −5.62444340096984547499514349297, −4.36613016059790117431481268962, −3.85954945220570202239268282076, −2.52546759988762890235560628809, −1.58047377832208132879145135158,
0.20242335877160520378777153231, 1.19740224496124865799795682913, 2.80361730866228347215185681162, 3.79539801083419903922191582546, 4.31932885452350448110337720336, 5.50800086112077364411533157711, 6.53315958928454127094516272394, 6.99755487372332821085314583344, 8.193522884237732563780669292464, 8.537703708009897172797438509830
