| L(s) = 1 | + (0.909 − 0.415i)2-s + (−0.880 − 2.99i)3-s + (0.654 − 0.755i)4-s + (−2.18 − 0.473i)5-s + (−2.04 − 2.36i)6-s + (3.80 − 0.547i)7-s + (0.281 − 0.959i)8-s + (−5.68 + 3.65i)9-s + (−2.18 + 0.477i)10-s + (−1.34 + 2.95i)11-s + (−2.84 − 1.29i)12-s + (−1.88 − 0.271i)13-s + (3.23 − 2.07i)14-s + (0.504 + 6.96i)15-s + (−0.142 − 0.989i)16-s + (4.55 − 3.94i)17-s + ⋯ |
| L(s) = 1 | + (0.643 − 0.293i)2-s + (−0.508 − 1.73i)3-s + (0.327 − 0.377i)4-s + (−0.977 − 0.211i)5-s + (−0.835 − 0.963i)6-s + (1.43 − 0.206i)7-s + (0.0996 − 0.339i)8-s + (−1.89 + 1.21i)9-s + (−0.690 + 0.150i)10-s + (−0.406 + 0.889i)11-s + (−0.820 − 0.374i)12-s + (−0.524 − 0.0753i)13-s + (0.864 − 0.555i)14-s + (0.130 + 1.79i)15-s + (−0.0355 − 0.247i)16-s + (1.10 − 0.956i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.458582 - 1.27349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.458582 - 1.27349i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 + 0.415i)T \) |
| 5 | \( 1 + (2.18 + 0.473i)T \) |
| 23 | \( 1 + (4.49 + 1.66i)T \) |
| good | 3 | \( 1 + (0.880 + 2.99i)T + (-2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-3.80 + 0.547i)T + (6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.34 - 2.95i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (1.88 + 0.271i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.55 + 3.94i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.42 + 2.79i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-0.494 - 0.571i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-9.29 - 2.72i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (0.639 + 0.995i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (3.65 + 2.35i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (0.624 + 2.12i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 3.45iT - 47T^{2} \) |
| 53 | \( 1 + (-10.6 + 1.52i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.24 - 8.63i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-3.29 - 0.967i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (10.2 - 4.67i)T + (43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.37 - 5.20i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (1.37 + 1.19i)T + (10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.926 - 6.44i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-0.361 - 0.562i)T + (-34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (2.31 - 0.680i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (0.0374 - 0.0582i)T + (-40.2 - 88.2i)T^{2} \) |
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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
−11.90600721811511657709560080216, −11.56332048159756522273744800406, −10.33298392869617694038294250682, −8.343861785361920093011054135183, −7.53608297099470995524426116198, −7.05426692212142137341294643783, −5.41081316852222157256793440487, −4.64188409642443056640278645859, −2.54942500918165262358041040509, −1.08053110790576304333905922391,
3.28478657444387273253587511229, 4.23679678290458166914515364448, 5.10618207166898566514125697395, 5.95600666483585671739984226140, 7.898836063878301018725638068303, 8.392386498804895763658433093375, 10.02333436763164183465833066553, 10.77821355013280576678058474230, 11.74455661300689703222134707747, 11.96471487305539451871070758765
