L(s) = 1 | + 1.69i·3-s + (0.589 + 2.15i)5-s + 4.22i·7-s + 0.118·9-s − 4.59·11-s − 0.978i·13-s + (−3.66 + 1.00i)15-s + 3.04i·17-s + 1.91·19-s − 7.17·21-s − i·23-s + (−4.30 + 2.54i)25-s + 5.29i·27-s + 0.737·29-s − 2.97·31-s + ⋯ |
L(s) = 1 | + 0.980i·3-s + (0.263 + 0.964i)5-s + 1.59i·7-s + 0.0394·9-s − 1.38·11-s − 0.271i·13-s + (−0.945 + 0.258i)15-s + 0.737i·17-s + 0.439·19-s − 1.56·21-s − 0.208i·23-s + (−0.861 + 0.508i)25-s + 1.01i·27-s + 0.136·29-s − 0.535·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.378300429\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.378300429\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.589 - 2.15i)T \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 - 1.69iT - 3T^{2} \) |
| 7 | \( 1 - 4.22iT - 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 + 0.978iT - 13T^{2} \) |
| 17 | \( 1 - 3.04iT - 17T^{2} \) |
| 19 | \( 1 - 1.91T + 19T^{2} \) |
| 29 | \( 1 - 0.737T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 + 8.93iT - 37T^{2} \) |
| 41 | \( 1 - 9.08T + 41T^{2} \) |
| 43 | \( 1 - 6.97iT - 43T^{2} \) |
| 47 | \( 1 - 2.58iT - 47T^{2} \) |
| 53 | \( 1 + 2.71iT - 53T^{2} \) |
| 59 | \( 1 - 7.13T + 59T^{2} \) |
| 61 | \( 1 + 0.731T + 61T^{2} \) |
| 67 | \( 1 + 7.16iT - 67T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 + 5.96iT - 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 + 4.39iT - 83T^{2} \) |
| 89 | \( 1 - 7.25T + 89T^{2} \) |
| 97 | \( 1 - 7.04iT - 97T^{2} \) |
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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
−9.605274441021746060348502285955, −9.168509690607417159499420746703, −8.127268254294846719703768290369, −7.44206149498478323881820828972, −6.23290662005010026682669885447, −5.59755380857213274316718754056, −4.96128996197293297936195974436, −3.74470977841866812323834711538, −2.83853571354908471913939707852, −2.13570576907331844946110549833,
0.51711774516158242238154182877, 1.35551642054243361834376659173, 2.53304662729709940825270138354, 3.91089529313654393787047838095, 4.76271999467242007053155208709, 5.53684760772099771251732206924, 6.66473373015401935948888817677, 7.40356515038382863118389366056, 7.78158666208116299446685617388, 8.643621771771817509728213437317