L(s) = 1 | + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)20-s + (−0.809 − 0.587i)25-s + (−0.618 − 1.90i)31-s + (0.809 + 0.587i)36-s − 0.999·45-s + (−0.309 + 0.951i)49-s + (−1.61 + 1.17i)59-s + (−0.309 − 0.951i)64-s + (0.618 − 1.90i)71-s + (0.809 + 0.587i)80-s + (−0.809 + 0.587i)81-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)20-s + (−0.809 − 0.587i)25-s + (−0.618 − 1.90i)31-s + (0.809 + 0.587i)36-s − 0.999·45-s + (−0.309 + 0.951i)49-s + (−1.61 + 1.17i)59-s + (−0.309 − 0.951i)64-s + (0.618 − 1.90i)71-s + (0.809 + 0.587i)80-s + (−0.809 + 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.043642640\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.043642640\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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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
−10.92578666281152917750732850494, −10.25881851021832487604136661434, −9.412595617578956705496853738427, −7.88473133727794176386653643295, −7.40935903978838611697412720426, −6.43228875898252587231759117997, −5.61084689896311571705213913272, −4.33391648754884933245420015259, −2.93444875780730462281345594157, −1.93699773126312220587693635294,
1.54717661651294741362798882938, 3.19098838552227910320030886240, 4.09347618665693302645786346977, 5.31204243810694259474995433068, 6.49138027950802718578027521651, 7.24984748089888624073211011470, 8.250934098712135693965273501209, 8.938020424124444982098554098999, 9.908577561828789285357601704170, 11.00714249449202219913876527928