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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.609862415\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.609862415\) |
\(L(1)\) |
|
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
−9.905047453160838557555947178868, −9.527308881005127497001650019730, −8.874171836849939427887581816672, −7.918725979961247470488906846370, −6.88928402697298387294447091967, −6.15758523147932605603450862004, −5.01908424709103718454780659728, −3.73845839731350988791793931379, −2.59903864581965609251739485447, −2.11120570570552100753675427989,
2.11120570570552100753675427989, 2.59903864581965609251739485447, 3.73845839731350988791793931379, 5.01908424709103718454780659728, 6.15758523147932605603450862004, 6.88928402697298387294447091967, 7.918725979961247470488906846370, 8.874171836849939427887581816672, 9.527308881005127497001650019730, 9.905047453160838557555947178868