| L(s) = 1 | + (1.11 + 0.808i)2-s + (−0.749 − 1.80i)3-s + (−0.0330 − 0.101i)4-s + (−2.04 + 0.902i)5-s + (0.629 − 2.62i)6-s + (1.71 − 2.79i)7-s + (0.895 − 2.75i)8-s + (−0.592 + 0.592i)9-s + (−3.00 − 0.649i)10-s + (2.06 − 2.42i)11-s + (−0.159 + 0.136i)12-s + (−1.27 + 5.32i)13-s + (4.16 − 1.72i)14-s + (3.16 + 3.02i)15-s + (3.05 − 2.21i)16-s + (−2.21 − 1.89i)17-s + ⋯ |
| L(s) = 1 | + (0.787 + 0.571i)2-s + (−0.432 − 1.04i)3-s + (−0.0165 − 0.0508i)4-s + (−0.914 + 0.403i)5-s + (0.256 − 1.07i)6-s + (0.646 − 1.05i)7-s + (0.316 − 0.974i)8-s + (−0.197 + 0.197i)9-s + (−0.950 − 0.205i)10-s + (0.623 − 0.730i)11-s + (−0.0459 + 0.0392i)12-s + (−0.354 + 1.47i)13-s + (1.11 − 0.460i)14-s + (0.817 + 0.781i)15-s + (0.763 − 0.554i)16-s + (−0.537 − 0.458i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20068 - 0.698659i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20068 - 0.698659i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.04 - 0.902i)T \) |
| 41 | \( 1 + (-5.01 - 3.98i)T \) |
| good | 2 | \( 1 + (-1.11 - 0.808i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.749 + 1.80i)T + (-2.12 + 2.12i)T^{2} \) |
| 7 | \( 1 + (-1.71 + 2.79i)T + (-3.17 - 6.23i)T^{2} \) |
| 11 | \( 1 + (-2.06 + 2.42i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (1.27 - 5.32i)T + (-11.5 - 5.90i)T^{2} \) |
| 17 | \( 1 + (2.21 + 1.89i)T + (2.65 + 16.7i)T^{2} \) |
| 19 | \( 1 + (0.294 - 0.481i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-0.797 - 5.03i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.762 - 9.69i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (-8.62 - 2.80i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.36 + 4.63i)T + (-21.7 - 29.9i)T^{2} \) |
| 43 | \( 1 + (-5.27 + 7.26i)T + (-13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (0.842 - 0.516i)T + (21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (0.592 + 0.693i)T + (-8.29 + 52.3i)T^{2} \) |
| 59 | \( 1 + (0.478 - 0.659i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.373 - 2.35i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (2.72 + 0.214i)T + (66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (6.15 + 5.25i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + 14.8iT - 73T^{2} \) |
| 79 | \( 1 + (0.697 - 1.68i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (7.44 - 7.44i)T - 83iT^{2} \) |
| 89 | \( 1 + (-8.93 + 2.14i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (0.479 - 6.09i)T + (-95.8 - 15.1i)T^{2} \) |
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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
−12.26783286720678279660404509315, −11.52436623198172712208713026636, −10.70268009142110913332209745111, −9.137709993660633515167933100082, −7.54770951354481768953343153031, −7.02941242855624122810532394064, −6.29804577666836369371584525200, −4.71575134706173488204087170052, −3.81646831498499900552936168765, −1.15334258022109491000360151988,
2.63921920934116054117574369299, 4.26962755219858097348851561035, 4.62905526394271155047999585832, 5.76043248072152932199393414818, 7.82647577948510194501081326597, 8.556461503201289868411831818710, 9.875434717813128515331875438430, 10.99956181479645389742771767879, 11.68673168865879376350189816016, 12.38660185546513147090905595331
