L(s) = 1 | + (−0.5 − 1.53i)3-s + (−0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s + (−1.30 + 0.951i)9-s + (1.30 + 0.951i)12-s + (1.30 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)20-s + (−0.5 + 0.363i)23-s + (−0.809 + 0.587i)25-s + (0.809 + 0.587i)27-s − 1.61·31-s + (0.5 − 1.53i)36-s + (0.618 + 1.90i)37-s + (−1.30 − 0.951i)45-s + ⋯ |
L(s) = 1 | + (−0.5 − 1.53i)3-s + (−0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s + (−1.30 + 0.951i)9-s + (1.30 + 0.951i)12-s + (1.30 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)20-s + (−0.5 + 0.363i)23-s + (−0.809 + 0.587i)25-s + (0.809 + 0.587i)27-s − 1.61·31-s + (0.5 − 1.53i)36-s + (0.618 + 1.90i)37-s + (−1.30 − 0.951i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5850773065\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5850773065\) |
\(L(1)\) |
|
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 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + 1.61T + T^{2} \) |
| 37 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + 1.61T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
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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
−8.943320678249151531439665553214, −7.904954959997543504440702696046, −7.59399011605906321498000793185, −6.87188035954628426006658604305, −6.09610734709155416329620091288, −5.50854141887253134921519222548, −4.35607107288654731363559031903, −3.26969602703835898303644575095, −2.44307703411544158364593415019, −1.30970897464233954898907680355,
0.41880795327091694292112542613, 1.99769906516580572374757182180, 3.72640332782068751236616132585, 4.15698765680689766724295497770, 4.92489563549297446153589317363, 5.60822994751187481968775830012, 5.86985306168920562638443962681, 7.29536980223008830167067553665, 8.570393625774396909774787118449, 8.913305810075342521528053463930