L(s) = 1 | + 1.41·3-s − 1.41·5-s + 7-s + 1.00·9-s + 1.41·13-s − 2.00·15-s − 1.41·19-s + 1.41·21-s + 1.00·25-s − 1.41·35-s + 2.00·39-s − 1.41·45-s + 49-s − 2.00·57-s − 1.41·59-s − 1.41·61-s + 1.00·63-s − 2.00·65-s − 2·71-s + 1.41·75-s − 2·79-s − 0.999·81-s + 1.41·83-s + 1.41·91-s + 2.00·95-s + 1.41·101-s − 2.00·105-s + ⋯ |
L(s) = 1 | + 1.41·3-s − 1.41·5-s + 7-s + 1.00·9-s + 1.41·13-s − 2.00·15-s − 1.41·19-s + 1.41·21-s + 1.00·25-s − 1.41·35-s + 2.00·39-s − 1.41·45-s + 49-s − 2.00·57-s − 1.41·59-s − 1.41·61-s + 1.00·63-s − 2.00·65-s − 2·71-s + 1.41·75-s − 2·79-s − 0.999·81-s + 1.41·83-s + 1.41·91-s + 2.00·95-s + 1.41·101-s − 2.00·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.377070691\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.377070691\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 1.41T + T^{2} \) |
| 5 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 2T + T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.44660375672113490465912434070, −8.969697968756809860395976038882, −8.596667266846424619982833358831, −7.955671642138657226801464781656, −7.38088076323783862002257168116, −6.10923970358682529950724072724, −4.53341792128439175629482969925, −3.95256151711235072160277779437, −3.04862959745114902768635689818, −1.69662029269896584039931522169,
1.69662029269896584039931522169, 3.04862959745114902768635689818, 3.95256151711235072160277779437, 4.53341792128439175629482969925, 6.10923970358682529950724072724, 7.38088076323783862002257168116, 7.955671642138657226801464781656, 8.596667266846424619982833358831, 8.969697968756809860395976038882, 10.44660375672113490465912434070