| L(s) = 1 | + (−0.213 + 1.39i)2-s + (−0.453 − 0.891i)3-s + (−1.90 − 0.595i)4-s + (−1.90 − 1.17i)5-s + (1.34 − 0.444i)6-s + (2.16 + 2.16i)7-s + (1.23 − 2.54i)8-s + (−0.587 + 0.809i)9-s + (2.04 − 2.40i)10-s + (3.61 + 4.97i)11-s + (0.335 + 1.97i)12-s + (0.749 + 4.72i)13-s + (−3.48 + 2.56i)14-s + (−0.182 + 2.22i)15-s + (3.29 + 2.27i)16-s + (−3.51 − 1.79i)17-s + ⋯ |
| L(s) = 1 | + (−0.150 + 0.988i)2-s + (−0.262 − 0.514i)3-s + (−0.954 − 0.297i)4-s + (−0.850 − 0.525i)5-s + (0.548 − 0.181i)6-s + (0.817 + 0.817i)7-s + (0.438 − 0.898i)8-s + (−0.195 + 0.269i)9-s + (0.647 − 0.762i)10-s + (1.09 + 1.50i)11-s + (0.0969 + 0.569i)12-s + (0.207 + 1.31i)13-s + (−0.931 + 0.685i)14-s + (−0.0472 + 0.575i)15-s + (0.822 + 0.568i)16-s + (−0.852 − 0.434i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.597803 + 0.681454i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.597803 + 0.681454i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.213 - 1.39i)T \) |
| 3 | \( 1 + (0.453 + 0.891i)T \) |
| 5 | \( 1 + (1.90 + 1.17i)T \) |
| good | 7 | \( 1 + (-2.16 - 2.16i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.61 - 4.97i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.749 - 4.72i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (3.51 + 1.79i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.678 - 2.08i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.374 + 2.36i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.48 - 0.480i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-8.29 + 2.69i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.07 + 0.644i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (5.32 + 3.86i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (8.72 - 8.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.47 - 1.77i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (5.54 - 2.82i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-7.51 - 5.46i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.715 - 0.519i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.52 + 2.98i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-5.96 - 1.93i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.906 - 0.143i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-2.98 + 9.17i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (6.49 + 3.31i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.440 - 0.606i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (7.49 + 14.7i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.92622933783520909396713947214, −11.49581124672777106978700178414, −9.731207894690329044608235906132, −8.857337095625254096438407144207, −8.146980284548179414799052322290, −7.08165074742975329449741907622, −6.37587693313219871589894138828, −4.81863071896285746059889297703, −4.35695496631782734055814145873, −1.63525780494770293253423324989,
0.832275097244959988864644307014, 3.18548485677921229270203166600, 3.93287424726146880726554050495, 5.02622095833048444559571964883, 6.57751430266996472196061327645, 8.121191215684774381430415282294, 8.533604304403455418382415221183, 9.981371567835757129001694221221, 10.85442539570461976414029949116, 11.26998214177748440318216584446
