| L(s) = 1 | + i·3-s + (−2.22 + 0.237i)5-s − 1.39i·7-s − 9-s + 1.82·11-s − 1.45i·13-s + (−0.237 − 2.22i)15-s + 6.33i·17-s + 0.666·19-s + 1.39·21-s − 3.19i·23-s + (4.88 − 1.05i)25-s − i·27-s − 2.76·29-s − 3.01·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.994 + 0.106i)5-s − 0.525i·7-s − 0.333·9-s + 0.550·11-s − 0.404i·13-s + (−0.0613 − 0.574i)15-s + 1.53i·17-s + 0.152·19-s + 0.303·21-s − 0.666i·23-s + (0.977 − 0.211i)25-s − 0.192i·27-s − 0.513·29-s − 0.542·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.334408684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.334408684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.22 - 0.237i)T \) |
| 67 | \( 1 + iT \) |
| good | 7 | \( 1 + 1.39iT - 7T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 + 1.45iT - 13T^{2} \) |
| 17 | \( 1 - 6.33iT - 17T^{2} \) |
| 19 | \( 1 - 0.666T + 19T^{2} \) |
| 23 | \( 1 + 3.19iT - 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 + 3.01T + 31T^{2} \) |
| 37 | \( 1 + 2.72iT - 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 - 0.406iT - 43T^{2} \) |
| 47 | \( 1 + 10.1iT - 47T^{2} \) |
| 53 | \( 1 - 1.57iT - 53T^{2} \) |
| 59 | \( 1 + 3.25T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 73 | \( 1 - 6.44iT - 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 2.66iT - 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 13.0iT - 97T^{2} \) |
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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
−8.499373024493774095443009134806, −7.79410796826789422281453327286, −7.04493653646114230311894400225, −6.30917315513349444255075997824, −5.38337086627298180617942502485, −4.50241242709909738004467213285, −3.75928620146945454813844254981, −3.41267335214201542278831415236, −1.99591027257641533219428374063, −0.59553877323511280602514143384,
0.72805225854339651916144398276, 1.90658185085532508787021993270, 2.99262386943626140527473669875, 3.71633093234188324288042764475, 4.74395343838209263218211533485, 5.38259479067467847174582227344, 6.40495814701476143002799588309, 7.07736821299996862819613863446, 7.62837274268131999859425705520, 8.343218797774431077106268127555
