| L(s) = 1 | + 3-s − 0.475i·5-s − 4.55i·7-s + 9-s − 1.65i·11-s − 0.475i·15-s − 7.93·17-s + 2.60i·19-s − 4.55i·21-s − 3.81·23-s + 4.77·25-s + 27-s + 0.823·29-s + 1.79i·31-s − 1.65i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.212i·5-s − 1.72i·7-s + 0.333·9-s − 0.498i·11-s − 0.122i·15-s − 1.92·17-s + 0.597i·19-s − 0.994i·21-s − 0.794·23-s + 0.954·25-s + 0.192·27-s + 0.152·29-s + 0.321i·31-s − 0.287i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.212837832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212837832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.475iT - 5T^{2} \) |
| 7 | \( 1 + 4.55iT - 7T^{2} \) |
| 11 | \( 1 + 1.65iT - 11T^{2} \) |
| 17 | \( 1 + 7.93T + 17T^{2} \) |
| 19 | \( 1 - 2.60iT - 19T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 - 0.823T + 29T^{2} \) |
| 31 | \( 1 - 1.79iT - 31T^{2} \) |
| 37 | \( 1 + 1.87iT - 37T^{2} \) |
| 41 | \( 1 + 10.7iT - 41T^{2} \) |
| 43 | \( 1 - 9.89T + 43T^{2} \) |
| 47 | \( 1 + 7.29iT - 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 1.48iT - 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 3.32iT - 67T^{2} \) |
| 71 | \( 1 + 1.90iT - 71T^{2} \) |
| 73 | \( 1 - 6.14iT - 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 14.7iT - 83T^{2} \) |
| 89 | \( 1 + 7.30iT - 89T^{2} \) |
| 97 | \( 1 - 4.18iT - 97T^{2} \) |
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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
−8.138513078158366243614328086987, −7.33075831121869457316869407502, −6.84668967153964972906503390149, −6.03006888192446788127216792551, −4.84844029260958887317864664013, −4.14315201842946464115289658490, −3.65743555867213303083132682785, −2.51858643019396176845908284705, −1.44814282174146379196212477250, −0.29793656963070699983934098918,
1.72573455496222565240452253598, 2.53886254645834849219701867316, 2.99947826157638680585515349147, 4.43580632029928198525469032016, 4.80474433529879497451328947488, 6.05537572540190151151325666893, 6.41123639520788777146623665294, 7.38779963558676212973965891463, 8.172509473797339077251391025677, 8.919820605562435531420415013818
