L(s) = 1 | + (1.22 + 0.707i)5-s + (1.22 − 0.707i)11-s + 13-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)25-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s − 1.41i·41-s − 43-s + (−1.22 − 0.707i)47-s + 2·55-s + (1.22 + 0.707i)65-s + (0.5 + 0.866i)67-s − 1.41i·71-s + (0.5 + 0.866i)73-s + ⋯ |
L(s) = 1 | + (1.22 + 0.707i)5-s + (1.22 − 0.707i)11-s + 13-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)25-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s − 1.41i·41-s − 43-s + (−1.22 − 0.707i)47-s + 2·55-s + (1.22 + 0.707i)65-s + (0.5 + 0.866i)67-s − 1.41i·71-s + (0.5 + 0.866i)73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.765753930\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.765753930\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−8.778838252342251208226100578777, −8.245157701645594619010576784110, −7.07269490053752020366963863634, −6.35029311160322581954581122817, −6.05464373439526974414179292244, −5.21209064831300390378905128625, −3.89622480377167573362255900124, −3.38874110237557171302395999255, −2.16898883701288896554781900893, −1.38735657387229775615645375447,
1.32584255438721064839670523976, 1.89520071877361127533978749223, 3.17671985002054231632029758597, 4.22209425775612428703850884425, 4.93073349739182697894477832075, 5.74284353428467623929114088432, 6.51109265396773838243124495833, 6.95566824828564566732306195737, 8.232265134921934373854119286456, 8.809664460173835891977841812462