| L(s) = 1 | − 2-s + 1.41·3-s + 4-s − 1.41·6-s − 8-s + 1.00·9-s + 1.41·12-s + 16-s − 1.41·17-s − 1.00·18-s − 1.41·19-s − 1.41·24-s + 25-s − 32-s + 1.41·34-s + 1.00·36-s + 1.41·38-s + 1.41·41-s + 1.41·48-s − 50-s − 2.00·51-s − 2.00·57-s − 1.41·59-s + 64-s − 2·67-s − 1.41·68-s − 1.00·72-s + ⋯ |
| L(s) = 1 | − 2-s + 1.41·3-s + 4-s − 1.41·6-s − 8-s + 1.00·9-s + 1.41·12-s + 16-s − 1.41·17-s − 1.00·18-s − 1.41·19-s − 1.41·24-s + 25-s − 32-s + 1.41·34-s + 1.00·36-s + 1.41·38-s + 1.41·41-s + 1.41·48-s − 50-s − 2.00·51-s − 2.00·57-s − 1.41·59-s + 64-s − 2·67-s − 1.41·68-s − 1.00·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8077912574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8077912574\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.41T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−11.15767815449137423742214366434, −10.49256219553940935271304948290, −9.319089283845472391981756214243, −8.836505842614331460046430730780, −8.123806088468366138004013113671, −7.16453561690616187677470284519, −6.21258376867662354644552691581, −4.32698307489476940748726754712, −2.93752529136674474231738964203, −2.00986025874645470728683418307,
2.00986025874645470728683418307, 2.93752529136674474231738964203, 4.32698307489476940748726754712, 6.21258376867662354644552691581, 7.16453561690616187677470284519, 8.123806088468366138004013113671, 8.836505842614331460046430730780, 9.319089283845472391981756214243, 10.49256219553940935271304948290, 11.15767815449137423742214366434
