L(s) = 1 | + (−0.281 − 1.04i)2-s + (0.866 + 0.5i)3-s + (0.710 − 0.410i)4-s + (1.02 + 1.02i)5-s + (0.281 − 1.04i)6-s + (−1.22 − 2.34i)7-s + (−2.16 − 2.16i)8-s + (0.499 + 0.866i)9-s + (0.788 − 1.36i)10-s + (0.972 − 0.260i)11-s + 0.820·12-s + (3.05 − 1.91i)13-s + (−2.11 + 1.94i)14-s + (0.376 + 1.40i)15-s + (−0.842 + 1.45i)16-s + (2.37 + 4.10i)17-s + ⋯ |
L(s) = 1 | + (−0.198 − 0.741i)2-s + (0.499 + 0.288i)3-s + (0.355 − 0.205i)4-s + (0.459 + 0.459i)5-s + (0.114 − 0.428i)6-s + (−0.462 − 0.886i)7-s + (−0.765 − 0.765i)8-s + (0.166 + 0.288i)9-s + (0.249 − 0.432i)10-s + (0.293 − 0.0785i)11-s + 0.236·12-s + (0.847 − 0.530i)13-s + (−0.565 + 0.519i)14-s + (0.0970 + 0.362i)15-s + (−0.210 + 0.364i)16-s + (0.574 + 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.32983 - 0.808374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32983 - 0.808374i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (1.22 + 2.34i)T \) |
| 13 | \( 1 + (-3.05 + 1.91i)T \) |
good | 2 | \( 1 + (0.281 + 1.04i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (-1.02 - 1.02i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.972 + 0.260i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.37 - 4.10i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.391 - 1.46i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (0.337 + 0.194i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.30 - 7.44i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.92 + 2.92i)T + 31iT^{2} \) |
| 37 | \( 1 + (-9.77 + 2.61i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (8.64 - 2.31i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (8.28 - 4.78i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.78 - 4.78i)T - 47iT^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + (0.889 + 0.238i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (8.36 - 4.82i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.71 - 6.39i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (2.56 + 0.687i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-2.46 + 2.46i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.06T + 79T^{2} \) |
| 83 | \( 1 + (2.43 + 2.43i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.79 + 10.4i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (3.55 - 13.2i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.44804371753231505528104813252, −10.51602658344477116101444053127, −10.18757805700599259514706056673, −9.208294124592292407652433833934, −7.954635283011745653417503807683, −6.71020055232466383398156079851, −5.87125176182888351786144127259, −3.88392148030370736967138141278, −3.07224513918794230123460256850, −1.50259849362265783496924169749,
2.05268138488607926584246972654, 3.38298449351644260987298909639, 5.30764966715980240131170183998, 6.24750314848819365728306878609, 7.11845410964278972429325438991, 8.261939382409357767358016979382, 9.016321022258502641151413055100, 9.702135908420925755193378494907, 11.43988178504605063848156430113, 12.01134006015694929421275177249