L(s) = 1 | − 0.732·3-s − 3.46·5-s − 0.732·7-s − 2.46·9-s − 4.73·11-s + 1.46·13-s + 2.53·15-s − 17-s + 5.46·19-s + 0.535·21-s + 4.73·23-s + 6.99·25-s + 4·27-s + 3.46·29-s + 6.19·31-s + 3.46·33-s + 2.53·35-s − 11.4·37-s − 1.07·39-s − 6·41-s + 12.3·43-s + 8.53·45-s − 6.92·47-s − 6.46·49-s + 0.732·51-s + 0.928·53-s + 16.3·55-s + ⋯ |
L(s) = 1 | − 0.422·3-s − 1.54·5-s − 0.276·7-s − 0.821·9-s − 1.42·11-s + 0.406·13-s + 0.654·15-s − 0.242·17-s + 1.25·19-s + 0.116·21-s + 0.986·23-s + 1.39·25-s + 0.769·27-s + 0.643·29-s + 1.11·31-s + 0.603·33-s + 0.428·35-s − 1.88·37-s − 0.171·39-s − 0.937·41-s + 1.88·43-s + 1.27·45-s − 1.01·47-s − 0.923·49-s + 0.102·51-s + 0.127·53-s + 2.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6472259265\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6472259265\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 - 7.46T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 + 9.46T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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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
−10.06568055112947454578800609571, −8.770908234888405638304989285304, −8.220642794620788386503302674655, −7.44599551055561857195532579541, −6.62249290870410303000612414614, −5.40275666838476630385164389814, −4.80362808321825323764039736383, −3.49952373717517545902107802904, −2.82916408279423896461630018632, −0.60223030651800219820213822827,
0.60223030651800219820213822827, 2.82916408279423896461630018632, 3.49952373717517545902107802904, 4.80362808321825323764039736383, 5.40275666838476630385164389814, 6.62249290870410303000612414614, 7.44599551055561857195532579541, 8.220642794620788386503302674655, 8.770908234888405638304989285304, 10.06568055112947454578800609571