| L(s) = 1 | − 5-s − 0.471i·11-s − 3.01i·13-s + 5.79·17-s + 5.14i·19-s + 1.83i·23-s + 25-s − 3.94i·29-s − 3.95i·31-s − 2.74·37-s − 4.82·41-s + 5.86·43-s + 6.56·47-s + 2.80i·53-s + 0.471i·55-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.142i·11-s − 0.835i·13-s + 1.40·17-s + 1.18i·19-s + 0.382i·23-s + 0.200·25-s − 0.731i·29-s − 0.710i·31-s − 0.450·37-s − 0.752·41-s + 0.893·43-s + 0.957·47-s + 0.385i·53-s + 0.0636i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780110931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780110931\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 0.471iT - 11T^{2} \) |
| 13 | \( 1 + 3.01iT - 13T^{2} \) |
| 17 | \( 1 - 5.79T + 17T^{2} \) |
| 19 | \( 1 - 5.14iT - 19T^{2} \) |
| 23 | \( 1 - 1.83iT - 23T^{2} \) |
| 29 | \( 1 + 3.94iT - 29T^{2} \) |
| 31 | \( 1 + 3.95iT - 31T^{2} \) |
| 37 | \( 1 + 2.74T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 - 6.56T + 47T^{2} \) |
| 53 | \( 1 - 2.80iT - 53T^{2} \) |
| 59 | \( 1 + 5.98T + 59T^{2} \) |
| 61 | \( 1 + 1.15iT - 61T^{2} \) |
| 67 | \( 1 + 4.25T + 67T^{2} \) |
| 71 | \( 1 - 9.68iT - 71T^{2} \) |
| 73 | \( 1 + 11.5iT - 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 4.45T + 83T^{2} \) |
| 89 | \( 1 + 4.41T + 89T^{2} \) |
| 97 | \( 1 - 3.56iT - 97T^{2} \) |
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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
−7.84447287675979329806786915377, −7.21361477196269375476287529357, −6.13323725877078285911735369739, −5.71768917746118836637224170087, −5.00150172661915846784515984179, −3.99674015020094899393368382448, −3.48713561963765999515980764516, −2.69358572709514694346432087235, −1.55952692746533215731567345564, −0.57974939924754021019945768657,
0.76763232343447019691591995987, 1.77644591714147747238837352301, 2.82542230683087524076875559851, 3.50014421439849187707977911338, 4.34432963002013875020823504232, 4.99067273066535963202698718428, 5.68034217948600791765890331043, 6.64119276046277839934706939926, 7.09050559416827787784865172322, 7.74601950162410411034124027799
