| L(s) = 1 | + (0.989 − 0.142i)2-s + (−0.909 − 0.415i)3-s + (0.959 − 0.281i)4-s + (0.670 − 2.13i)5-s + (−0.959 − 0.281i)6-s + (0.0999 + 0.155i)7-s + (0.909 − 0.415i)8-s + (0.654 + 0.755i)9-s + (0.360 − 2.20i)10-s + (−0.591 + 4.11i)11-s + (−0.989 − 0.142i)12-s + (3.08 − 4.80i)13-s + (0.121 + 0.139i)14-s + (−1.49 + 1.66i)15-s + (0.841 − 0.540i)16-s + (1.78 − 6.08i)17-s + ⋯ |
| L(s) = 1 | + (0.699 − 0.100i)2-s + (−0.525 − 0.239i)3-s + (0.479 − 0.140i)4-s + (0.300 − 0.953i)5-s + (−0.391 − 0.115i)6-s + (0.0377 + 0.0588i)7-s + (0.321 − 0.146i)8-s + (0.218 + 0.251i)9-s + (0.113 − 0.697i)10-s + (−0.178 + 1.24i)11-s + (−0.285 − 0.0410i)12-s + (0.855 − 1.33i)13-s + (0.0323 + 0.0373i)14-s + (−0.386 + 0.429i)15-s + (0.210 − 0.135i)16-s + (0.433 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61316 - 1.26112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61316 - 1.26112i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.989 + 0.142i)T \) |
| 3 | \( 1 + (0.909 + 0.415i)T \) |
| 5 | \( 1 + (-0.670 + 2.13i)T \) |
| 23 | \( 1 + (4.79 - 0.0673i)T \) |
| good | 7 | \( 1 + (-0.0999 - 0.155i)T + (-2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.591 - 4.11i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-3.08 + 4.80i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.78 + 6.08i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-2.06 + 0.605i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-4.70 - 1.38i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.03 + 4.46i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (4.72 - 4.09i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (1.65 - 1.90i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (4.80 + 2.19i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 2.44iT - 47T^{2} \) |
| 53 | \( 1 + (-6.66 - 10.3i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-9.13 - 5.87i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (5.02 + 11.0i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-8.92 + 1.28i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.970 - 6.74i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-4.11 - 13.9i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-13.8 - 8.88i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (0.426 - 0.369i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (7.29 - 15.9i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-1.35 - 1.17i)T + (13.8 + 96.0i)T^{2} \) |
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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
−10.21074619919224018995792928728, −9.742861095358077402627754778923, −8.423518181758015747003445376917, −7.56552396977391569110678085687, −6.58490604720563927153450732128, −5.35580489023258990340819000438, −5.15850744835786649762125896420, −3.89223468524011248262890670921, −2.41483236761400515673837552106, −0.998065215860565823361396109916,
1.78356753604996210983794085267, 3.37231263706714865886574107743, 3.98718019125760576372665383441, 5.43272941466414226228645257209, 6.19109810371132761259457533465, 6.68098894507784699621752589598, 7.933465321004082969045006886071, 8.898518036950265765034466224455, 10.20549561129438045321374424950, 10.70919466372489524666690471176
