L(s) = 1 | + 5·13-s + 19-s − 5·25-s − 11·31-s + 11·37-s + 13·43-s + 14·61-s − 5·67-s + 17·73-s − 17·79-s + 14·97-s + 13·103-s − 19·109-s + ⋯ |
L(s) = 1 | + 1.38·13-s + 0.229·19-s − 25-s − 1.97·31-s + 1.80·37-s + 1.98·43-s + 1.79·61-s − 0.610·67-s + 1.98·73-s − 1.91·79-s + 1.42·97-s + 1.28·103-s − 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.107089649\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.107089649\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.84014573568306323019593552501, −7.39234135084387044471346727250, −6.42038961191263648676844914308, −5.86596953808878933509990679450, −5.26744708845361220498654117170, −4.09271816910079491119154184020, −3.77625661608191993170587269281, −2.71046287329663807898221422433, −1.77358275157103534777873174560, −0.75655752694590402311313676511,
0.75655752694590402311313676511, 1.77358275157103534777873174560, 2.71046287329663807898221422433, 3.77625661608191993170587269281, 4.09271816910079491119154184020, 5.26744708845361220498654117170, 5.86596953808878933509990679450, 6.42038961191263648676844914308, 7.39234135084387044471346727250, 7.84014573568306323019593552501