L(s) = 1 | + 2-s + 3-s + 4-s − 0.312·5-s + 6-s − 1.12·7-s + 8-s + 9-s − 0.312·10-s + 4.84·11-s + 12-s − 2.90·13-s − 1.12·14-s − 0.312·15-s + 16-s − 2.37·17-s + 18-s + 19-s − 0.312·20-s − 1.12·21-s + 4.84·22-s + 7.43·23-s + 24-s − 4.90·25-s − 2.90·26-s + 27-s − 1.12·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.139·5-s + 0.408·6-s − 0.426·7-s + 0.353·8-s + 0.333·9-s − 0.0986·10-s + 1.45·11-s + 0.288·12-s − 0.804·13-s − 0.301·14-s − 0.0805·15-s + 0.250·16-s − 0.575·17-s + 0.235·18-s + 0.229·19-s − 0.0697·20-s − 0.245·21-s + 1.03·22-s + 1.54·23-s + 0.204·24-s − 0.980·25-s − 0.568·26-s + 0.192·27-s − 0.213·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.081867756\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.081867756\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 + 0.312T + 5T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 11 | \( 1 - 4.84T + 11T^{2} \) |
| 13 | \( 1 + 2.90T + 13T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 - 2.54T + 29T^{2} \) |
| 31 | \( 1 + 1.67T + 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 - 9.40T + 47T^{2} \) |
| 59 | \( 1 - 1.03T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 3.92T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 + 2.83T + 79T^{2} \) |
| 83 | \( 1 - 9.84T + 83T^{2} \) |
| 89 | \( 1 + 4.79T + 89T^{2} \) |
| 97 | \( 1 - 4.44T + 97T^{2} \) |
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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
−7.931055122389247441576136292713, −7.16562111160319652441835655997, −6.76821635479800098691063557109, −5.96906013545899853877353816855, −5.09436934496963354694225284838, −4.24835512209240473840365178436, −3.75034053545048603217901198087, −2.87651861608429791113740918739, −2.11679605850612478447534576307, −0.953089253865494263116087313569,
0.953089253865494263116087313569, 2.11679605850612478447534576307, 2.87651861608429791113740918739, 3.75034053545048603217901198087, 4.24835512209240473840365178436, 5.09436934496963354694225284838, 5.96906013545899853877353816855, 6.76821635479800098691063557109, 7.16562111160319652441835655997, 7.931055122389247441576136292713