L(s) = 1 | + (0.309 − 0.951i)4-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)16-s + (1.61 − 1.17i)31-s + (−0.309 − 0.951i)36-s + (−0.809 − 0.587i)49-s + (−0.618 + 1.90i)59-s + (−0.809 + 0.587i)64-s + (−1.61 − 1.17i)71-s + (0.309 − 0.951i)81-s + 2·89-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)4-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)16-s + (1.61 − 1.17i)31-s + (−0.309 − 0.951i)36-s + (−0.809 − 0.587i)49-s + (−0.618 + 1.90i)59-s + (−0.809 + 0.587i)64-s + (−1.61 − 1.17i)71-s + (0.309 − 0.951i)81-s + 2·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.389961530\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.389961530\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 + (0.309 - 0.951i)T^{2} \) |
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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
−8.948599195229487476247271131885, −7.903078198723099393824615954269, −7.14909144222917486213540542846, −6.38688192114351780713308688419, −5.89237501676137510682339954913, −4.83279588719347990127203577688, −4.20946782737479113767621070635, −3.02747941243955528740200309123, −1.95125099815500254925731718884, −0.921513808326219850436467154319,
1.54933882242620815637663757873, 2.60517708748495607011610822641, 3.44184515103002527813404887507, 4.40127496899301648621571448866, 4.99569410438809019900586885819, 6.26215594049809338450300095306, 6.87372306665595397042354284148, 7.63654910972502878883701996790, 8.175375862722574562854153422966, 8.907463579617866558483013650831