L(s) = 1 | + (1.32 − 0.498i)2-s + (1.08 − 1.08i)3-s + (1.50 − 1.31i)4-s + (−0.398 + 2.20i)5-s + (0.898 − 1.98i)6-s + (−0.555 − 0.555i)7-s + (1.33 − 2.49i)8-s + 0.626i·9-s + (0.570 + 3.11i)10-s − 2.61i·11-s + (0.198 − 3.07i)12-s + (0.707 + 0.707i)13-s + (−1.01 − 0.457i)14-s + (1.96 + 2.83i)15-s + (0.515 − 3.96i)16-s + (−2.63 + 2.63i)17-s + ⋯ |
L(s) = 1 | + (0.935 − 0.352i)2-s + (0.628 − 0.628i)3-s + (0.751 − 0.659i)4-s + (−0.178 + 0.984i)5-s + (0.366 − 0.810i)6-s + (−0.209 − 0.209i)7-s + (0.470 − 0.882i)8-s + 0.208i·9-s + (0.180 + 0.983i)10-s − 0.787i·11-s + (0.0574 − 0.887i)12-s + (0.196 + 0.196i)13-s + (−0.270 − 0.122i)14-s + (0.506 + 0.730i)15-s + (0.128 − 0.991i)16-s + (−0.639 + 0.639i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.20477 - 0.928570i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.20477 - 0.928570i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.32 + 0.498i)T \) |
| 5 | \( 1 + (0.398 - 2.20i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-1.08 + 1.08i)T - 3iT^{2} \) |
| 7 | \( 1 + (0.555 + 0.555i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.61iT - 11T^{2} \) |
| 17 | \( 1 + (2.63 - 2.63i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 23 | \( 1 + (0.817 - 0.817i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.74iT - 29T^{2} \) |
| 31 | \( 1 - 1.39iT - 31T^{2} \) |
| 37 | \( 1 + (0.838 - 0.838i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.268T + 41T^{2} \) |
| 43 | \( 1 + (-3.82 + 3.82i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.05 - 8.05i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.61 - 6.61i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 9.92T + 61T^{2} \) |
| 67 | \( 1 + (4.22 + 4.22i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 + (7.67 + 7.67i)T + 73iT^{2} \) |
| 79 | \( 1 + 6.13T + 79T^{2} \) |
| 83 | \( 1 + (-2.27 + 2.27i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.29iT - 89T^{2} \) |
| 97 | \( 1 + (3.44 - 3.44i)T - 97iT^{2} \) |
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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
−12.00496224205893265420290075101, −10.78852661283748145555864930707, −10.58329139597755232168544253958, −8.888130130396203965964208482389, −7.70387274284183227074013983415, −6.75957915336789702114897729145, −5.94899506323147004908082235341, −4.23488970616909764341180176326, −3.11867645341281419737894860729, −2.02991622412712541318387865308,
2.43965517709263886111689577747, 3.96942781049828369551239794739, 4.58615329522924029987100812630, 5.85695152505659584801120595963, 7.07468243053418445440391930105, 8.336692436877610898822472255090, 9.024022861875974164100851388717, 10.12543014476426234427391453746, 11.44753298488909842508604577314, 12.40133455363252129053744531518