| L(s) = 1 | + (−0.306 − 1.38i)2-s + (−0.713 − 1.40i)3-s + (−1.81 + 0.846i)4-s + (−1.23 + 1.86i)5-s + (−1.71 + 1.41i)6-s + (−0.707 − 0.707i)7-s + (1.72 + 2.24i)8-s + (0.311 − 0.429i)9-s + (2.95 + 1.13i)10-s + (−0.0883 − 0.121i)11-s + (2.47 + 1.93i)12-s + (0.899 + 5.67i)13-s + (−0.759 + 1.19i)14-s + (3.49 + 0.399i)15-s + (2.56 − 3.06i)16-s + (4.70 + 2.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.216 − 0.976i)2-s + (−0.411 − 0.808i)3-s + (−0.906 + 0.423i)4-s + (−0.552 + 0.833i)5-s + (−0.699 + 0.577i)6-s + (−0.267 − 0.267i)7-s + (0.609 + 0.792i)8-s + (0.103 − 0.143i)9-s + (0.933 + 0.358i)10-s + (−0.0266 − 0.0366i)11-s + (0.715 + 0.558i)12-s + (0.249 + 1.57i)13-s + (−0.203 + 0.318i)14-s + (0.901 + 0.103i)15-s + (0.642 − 0.766i)16-s + (1.14 + 0.581i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.795608 - 0.476413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.795608 - 0.476413i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.306 + 1.38i)T \) |
| 5 | \( 1 + (1.23 - 1.86i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| good | 3 | \( 1 + (0.713 + 1.40i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (0.0883 + 0.121i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.899 - 5.67i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-4.70 - 2.39i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (0.209 + 0.644i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.420 + 2.65i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-0.763 - 0.248i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.71 + 2.18i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.42 + 0.542i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-4.00 - 2.90i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (3.01 - 3.01i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.73 + 2.91i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.0497 + 0.0253i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-8.65 - 6.28i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-12.2 + 8.91i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (3.58 - 7.04i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-6.06 - 1.97i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (9.06 + 1.43i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (0.750 - 2.31i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (6.15 + 3.13i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (2.62 + 3.61i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.186 - 0.366i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.34847810667321282522970965452, −9.738697701822842311636087819051, −8.599119202519490501090178777734, −7.66806109762689792237646407195, −6.86322772942829059473147041418, −6.04721468796955651937066587072, −4.38881345045974447632691275853, −3.62463447707573306657415724296, −2.35342432578925418949732329622, −0.977245694037742265347129494962,
0.801615496115459018524168948790, 3.40698132106505651333728286882, 4.46716202379894876903419115452, 5.34718691122110456368401174410, 5.77536090072863249699857909948, 7.30294966788689519795062185727, 7.993491580742037096124347134722, 8.747563907024399215621378453113, 9.824202403007480796165787399831, 10.17801293186230370491714311626
