L(s) = 1 | − 1.61i·3-s − i·7-s − 1.61·9-s + i·11-s + 1.61·13-s + 17-s − 1.61i·19-s − 1.61·21-s + i·27-s − 29-s − 0.618i·31-s + 1.61·33-s − 2.61i·39-s + 41-s − i·43-s + ⋯ |
L(s) = 1 | − 1.61i·3-s − i·7-s − 1.61·9-s + i·11-s + 1.61·13-s + 17-s − 1.61i·19-s − 1.61·21-s + i·27-s − 29-s − 0.618i·31-s + 1.61·33-s − 2.61i·39-s + 41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.350073088\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.350073088\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 1.61iT - T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 - 1.61T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + 1.61iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + 0.618iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 - 0.618iT - T^{2} \) |
| 53 | \( 1 + 0.618T + T^{2} \) |
| 59 | \( 1 - 0.618iT - T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 + 0.618iT - T^{2} \) |
| 71 | \( 1 - iT - T^{2} \) |
| 73 | \( 1 + 0.618T + T^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 0.618T + T^{2} \) |
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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
−8.087781238136283289135621862359, −7.39737675226725226777886458798, −7.13903009919667631776509442086, −6.29911330210018601226373462263, −5.67213465145316684833732265330, −4.52302139786196214838776896815, −3.65894834259825561455004339511, −2.61148002864408513649596611318, −1.58655329027364066929084921440, −0.839094865532504771603129335201,
1.53465440101401524412431320365, 3.09020815438981819715662561236, 3.48322579645655121941470518930, 4.21910054316404716766348125082, 5.30534968837143397432003601201, 5.83183253709136063028482332506, 6.19498726305726455646597250326, 7.81195743629333537415758356538, 8.355578984449951290210364343077, 9.049031713650509966180474118448