| L(s) = 1 | − 2.21i·3-s + 2.69i·5-s − 2.88i·7-s − 1.90·9-s − 5.22i·11-s − 3.34·13-s + 5.95·15-s + (−3.60 − 1.99i)17-s + 2.82·19-s − 6.39·21-s − 0.998i·23-s − 2.23·25-s − 2.42i·27-s − 3.66i·29-s + 4.60i·31-s + ⋯ |
| L(s) = 1 | − 1.27i·3-s + 1.20i·5-s − 1.09i·7-s − 0.635·9-s − 1.57i·11-s − 0.926·13-s + 1.53·15-s + (−0.874 − 0.484i)17-s + 0.648·19-s − 1.39·21-s − 0.208i·23-s − 0.447·25-s − 0.465i·27-s − 0.680i·29-s + 0.827i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8056415037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8056415037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.60 + 1.99i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.21iT - 3T^{2} \) |
| 5 | \( 1 - 2.69iT - 5T^{2} \) |
| 7 | \( 1 + 2.88iT - 7T^{2} \) |
| 11 | \( 1 + 5.22iT - 11T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 0.998iT - 23T^{2} \) |
| 29 | \( 1 + 3.66iT - 29T^{2} \) |
| 31 | \( 1 - 4.60iT - 31T^{2} \) |
| 37 | \( 1 + 3.04iT - 37T^{2} \) |
| 41 | \( 1 - 6.46iT - 41T^{2} \) |
| 43 | \( 1 + 5.39T + 43T^{2} \) |
| 47 | \( 1 - 2.80T + 47T^{2} \) |
| 53 | \( 1 + 6.70T + 53T^{2} \) |
| 61 | \( 1 - 9.33iT - 61T^{2} \) |
| 67 | \( 1 - 0.00861T + 67T^{2} \) |
| 71 | \( 1 - 2.93iT - 71T^{2} \) |
| 73 | \( 1 + 7.83iT - 73T^{2} \) |
| 79 | \( 1 - 5.00iT - 79T^{2} \) |
| 83 | \( 1 - 6.57T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 3.82iT - 97T^{2} \) |
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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
−7.70469491386575164362534876635, −7.29102393131564373096115954070, −6.63046296100712813033075210928, −6.23971566049124293385958770757, −5.13396256699357015599073646715, −4.03338404430658856177102929241, −3.06456851217544831567565456172, −2.50465761849737034568273766160, −1.23504125896233380495262099726, −0.23256685205457937832466738325,
1.66385641930228023798838459198, 2.51820337133058313135487708563, 3.74533881222688035792100956362, 4.58146651430693265587171838871, 4.98196028305506582062758442387, 5.47015191394253966887827550945, 6.62730624858742732783610348894, 7.53200423396594078179897447017, 8.385291371423484286976631988976, 9.212744748437763594117280056442
