L(s) = 1 | − 2.45·2-s + 3-s + 4.00·4-s + 4.42·5-s − 2.45·6-s + 1.04·7-s − 4.91·8-s + 9-s − 10.8·10-s + 11-s + 4.00·12-s − 2.55·14-s + 4.42·15-s + 4.03·16-s + 1.91·17-s − 2.45·18-s − 2.86·19-s + 17.7·20-s + 1.04·21-s − 2.45·22-s − 2.43·23-s − 4.91·24-s + 14.5·25-s + 27-s + 4.17·28-s − 7.02·29-s − 10.8·30-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 0.577·3-s + 2.00·4-s + 1.97·5-s − 1.00·6-s + 0.393·7-s − 1.73·8-s + 0.333·9-s − 3.42·10-s + 0.301·11-s + 1.15·12-s − 0.682·14-s + 1.14·15-s + 1.00·16-s + 0.464·17-s − 0.577·18-s − 0.658·19-s + 3.95·20-s + 0.227·21-s − 0.522·22-s − 0.506·23-s − 1.00·24-s + 2.90·25-s + 0.192·27-s + 0.789·28-s − 1.30·29-s − 1.97·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.844441295\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.844441295\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 5 | \( 1 - 4.42T + 5T^{2} \) |
| 7 | \( 1 - 1.04T + 7T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 - 8.30T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 + 6.12T + 43T^{2} \) |
| 47 | \( 1 - 8.92T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 2.69T + 61T^{2} \) |
| 67 | \( 1 + 7.53T + 67T^{2} \) |
| 71 | \( 1 - 2.65T + 71T^{2} \) |
| 73 | \( 1 + 3.87T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 - 5.54T + 83T^{2} \) |
| 89 | \( 1 - 9.19T + 89T^{2} \) |
| 97 | \( 1 - 0.785T + 97T^{2} \) |
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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
−8.279963037920507689489932887056, −7.74005327618980820312085314659, −6.80643913819230916493052536292, −6.29582741763634376293933312358, −5.59198458684649714091896060906, −4.57330514557006212673098198650, −3.17981414679422297333456161768, −2.18617795276830509954816799400, −1.86577811827156771308689382576, −0.959477417927984261982753351920,
0.959477417927984261982753351920, 1.86577811827156771308689382576, 2.18617795276830509954816799400, 3.17981414679422297333456161768, 4.57330514557006212673098198650, 5.59198458684649714091896060906, 6.29582741763634376293933312358, 6.80643913819230916493052536292, 7.74005327618980820312085314659, 8.279963037920507689489932887056