| L(s) = 1 | + 5-s − 4.35i·7-s + 0.0533·11-s − 4.90·13-s + 1.18·17-s + 5.30i·19-s + (−1.90 − 4.40i)23-s + 25-s + 3.23i·29-s − 6.80·31-s − 4.35i·35-s + 4.86i·37-s − 3.15i·41-s − 10.5i·43-s − 11.6i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.64i·7-s + 0.0160·11-s − 1.36·13-s + 0.286·17-s + 1.21i·19-s + (−0.397 − 0.917i)23-s + 0.200·25-s + 0.600i·29-s − 1.22·31-s − 0.736i·35-s + 0.799i·37-s − 0.493i·41-s − 1.60i·43-s − 1.70i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3907006343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3907006343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + (1.90 + 4.40i)T \) |
| good | 7 | \( 1 + 4.35iT - 7T^{2} \) |
| 11 | \( 1 - 0.0533T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 19 | \( 1 - 5.30iT - 19T^{2} \) |
| 29 | \( 1 - 3.23iT - 29T^{2} \) |
| 31 | \( 1 + 6.80T + 31T^{2} \) |
| 37 | \( 1 - 4.86iT - 37T^{2} \) |
| 41 | \( 1 + 3.15iT - 41T^{2} \) |
| 43 | \( 1 + 10.5iT - 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 + 6.20T + 53T^{2} \) |
| 59 | \( 1 - 4.11iT - 59T^{2} \) |
| 61 | \( 1 - 0.265iT - 61T^{2} \) |
| 67 | \( 1 - 12.6iT - 67T^{2} \) |
| 71 | \( 1 - 1.13iT - 71T^{2} \) |
| 73 | \( 1 - 9.67T + 73T^{2} \) |
| 79 | \( 1 - 3.25iT - 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 6.81iT - 97T^{2} \) |
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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
−7.938003707404662231381416550504, −7.11920293706809244872804534742, −6.88987494664466071118048689554, −5.72801604840882932517019716883, −5.04540826791550773382267467202, −4.13457850009514339803610314600, −3.53885457564965017304370622511, −2.34529646767871989561175595147, −1.36070901118792349873714478452, −0.10350270553510140017622294236,
1.69919604391917271772931250485, 2.51662729640643998242545677577, 3.10925544042766414995929853391, 4.50757043218714070217211068837, 5.18624960813786612205147435325, 5.78909541654989715023366935637, 6.47652075909664952494551985900, 7.44569408595316878371825573482, 8.034761730206224380512534867393, 9.009531649826266587644952299235
