L(s) = 1 | + (0.192 − 0.265i)2-s + (−0.914 − 0.297i)3-s + (0.584 + 1.79i)4-s + (1.74 + 1.39i)5-s + (−0.255 + 0.185i)6-s + (3.11 − 1.01i)7-s + (1.21 + 0.394i)8-s + (−1.67 − 1.21i)9-s + (0.708 − 0.194i)10-s − 1.82i·12-s + (3.06 − 4.21i)13-s + (0.332 − 1.02i)14-s + (−1.18 − 1.79i)15-s + (−2.72 + 1.97i)16-s + (2.11 + 2.91i)17-s + (−0.647 + 0.210i)18-s + ⋯ |
L(s) = 1 | + (0.136 − 0.187i)2-s + (−0.528 − 0.171i)3-s + (0.292 + 0.899i)4-s + (0.780 + 0.624i)5-s + (−0.104 + 0.0757i)6-s + (1.17 − 0.382i)7-s + (0.429 + 0.139i)8-s + (−0.559 − 0.406i)9-s + (0.223 − 0.0613i)10-s − 0.525i·12-s + (0.848 − 1.16i)13-s + (0.0887 − 0.273i)14-s + (−0.305 − 0.464i)15-s + (−0.680 + 0.494i)16-s + (0.513 + 0.706i)17-s + (−0.152 + 0.0496i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78697 + 0.368533i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78697 + 0.368533i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.74 - 1.39i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.192 + 0.265i)T + (-0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.914 + 0.297i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (-3.11 + 1.01i)T + (5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.06 + 4.21i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.11 - 2.91i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.54 - 4.76i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 5.84iT - 23T^{2} \) |
| 29 | \( 1 + (-0.632 - 1.94i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.63 - 1.91i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (5.93 - 1.92i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.15 + 9.70i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 0.596iT - 43T^{2} \) |
| 47 | \( 1 + (0.914 + 0.297i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.231 - 0.318i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.162 - 0.501i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.30 + 3.13i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 8.45iT - 67T^{2} \) |
| 71 | \( 1 + (10.1 - 7.36i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.09 + 2.63i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.29 - 1.66i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (7.13 + 9.82i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (-6.63 + 9.13i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−10.65038783053426940335733026781, −10.40798731870909582863502438715, −8.617337026751197976731690464395, −8.177343788803139690264422292878, −7.08541826464207548303972199922, −6.14684109090820714422497692094, −5.35223713772089509248293930598, −3.92701105089497352698106240441, −2.91830626118158266055740263483, −1.53296521010062146535464745044,
1.27628382362553213690983204388, 2.31365068015476425484222986227, 4.56027716822357224217315560462, 5.14886898475127810930473305786, 5.84124837656518530076337194423, 6.70261403440175119370055101670, 8.030176361867603932987998276347, 9.002121673498855538642938680557, 9.675359894214167869648966617379, 10.79338341737990204398061377087