| L(s) = 1 | + (−0.876 − 1.01i)2-s + (−0.658 + 1.24i)3-s + (0.0287 − 0.190i)4-s + (−1.55 + 1.60i)5-s + (1.84 − 0.421i)6-s + (1.11 − 2.40i)7-s + (−2.49 + 1.56i)8-s + (0.572 + 0.839i)9-s + (3.00 + 0.170i)10-s + (−4.78 − 3.26i)11-s + (0.218 + 0.161i)12-s + (−1.88 + 5.37i)13-s + (−3.42 + 0.970i)14-s + (−0.982 − 2.99i)15-s + (3.41 + 1.05i)16-s + (−5.54 + 2.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.620 − 0.720i)2-s + (−0.380 + 0.719i)3-s + (0.0143 − 0.0953i)4-s + (−0.694 + 0.719i)5-s + (0.753 − 0.172i)6-s + (0.420 − 0.907i)7-s + (−0.882 + 0.554i)8-s + (0.190 + 0.279i)9-s + (0.949 + 0.0539i)10-s + (−1.44 − 0.984i)11-s + (0.0631 + 0.0465i)12-s + (−0.521 + 1.49i)13-s + (−0.914 + 0.259i)14-s + (−0.253 − 0.772i)15-s + (0.854 + 0.263i)16-s + (−1.34 + 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0815648 + 0.167290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0815648 + 0.167290i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.55 - 1.60i)T \) |
| 7 | \( 1 + (-1.11 + 2.40i)T \) |
| good | 2 | \( 1 + (0.876 + 1.01i)T + (-0.298 + 1.97i)T^{2} \) |
| 3 | \( 1 + (0.658 - 1.24i)T + (-1.68 - 2.47i)T^{2} \) |
| 11 | \( 1 + (4.78 + 3.26i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (1.88 - 5.37i)T + (-10.1 - 8.10i)T^{2} \) |
| 17 | \( 1 + (5.54 - 2.42i)T + (11.5 - 12.4i)T^{2} \) |
| 19 | \( 1 + (1.06 - 1.83i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.622 - 1.42i)T + (-15.6 - 16.8i)T^{2} \) |
| 29 | \( 1 + (-2.59 + 2.07i)T + (6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (0.182 - 0.105i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.83 - 5.20i)T + (-10.9 - 35.3i)T^{2} \) |
| 41 | \( 1 + (-2.31 - 0.529i)T + (36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.509 + 0.810i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (-7.84 + 6.75i)T + (7.00 - 46.4i)T^{2} \) |
| 53 | \( 1 + (2.63 + 3.57i)T + (-15.6 + 50.6i)T^{2} \) |
| 59 | \( 1 + (0.472 - 0.438i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (0.465 + 3.09i)T + (-58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-2.73 - 0.734i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (4.68 - 5.87i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (5.78 + 4.97i)T + (10.8 + 72.1i)T^{2} \) |
| 79 | \( 1 + (-5.34 - 3.08i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.48 + 2.96i)T + (64.8 - 51.7i)T^{2} \) |
| 89 | \( 1 + (3.40 - 2.32i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (-0.434 - 0.434i)T + 97iT^{2} \) |
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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
−11.80412784133948657869778435243, −11.11483782147131346799795869351, −10.60017946572489452003699450928, −10.06137361326725069193890602189, −8.694194305219531350231210352360, −7.69925955062069309551361383266, −6.44952380197166822439786774126, −4.96228848591701978810653540508, −3.88714599076187381547055626698, −2.24336032727713197211360468674,
0.17716513802723996551446221340, 2.62306530194524855182668267406, 4.68986892140843227830210789653, 5.74381005069971420705136681525, 7.14545573538771903003154313154, 7.69707690022001051257680956830, 8.546616385671815757727995567577, 9.457199898116777804438227803871, 10.86469097534719993477319545584, 12.16855065634288106620653791942
