L(s) = 1 | + (−0.733 − 0.680i)3-s + (0.623 − 0.781i)4-s + (0.0747 − 0.997i)7-s + (0.0747 + 0.997i)9-s + (−0.988 + 0.149i)12-s + (0.365 − 0.930i)13-s + (−0.222 − 0.974i)16-s + (−0.0747 − 0.129i)19-s + (−0.733 + 0.680i)21-s + (0.0747 + 0.997i)25-s + (0.623 − 0.781i)27-s + (−0.733 − 0.680i)28-s + (−0.623 − 1.07i)31-s + (0.826 + 0.563i)36-s + (0.455 + 0.571i)37-s + ⋯ |
L(s) = 1 | + (−0.733 − 0.680i)3-s + (0.623 − 0.781i)4-s + (0.0747 − 0.997i)7-s + (0.0747 + 0.997i)9-s + (−0.988 + 0.149i)12-s + (0.365 − 0.930i)13-s + (−0.222 − 0.974i)16-s + (−0.0747 − 0.129i)19-s + (−0.733 + 0.680i)21-s + (0.0747 + 0.997i)25-s + (0.623 − 0.781i)27-s + (−0.733 − 0.680i)28-s + (−0.623 − 1.07i)31-s + (0.826 + 0.563i)36-s + (0.455 + 0.571i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9928353546\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9928353546\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.733 + 0.680i)T \) |
| 7 | \( 1 + (-0.0747 + 0.997i)T \) |
| 13 | \( 1 + (-0.365 + 0.930i)T \) |
good | 2 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 5 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 11 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 17 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 19 | \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 29 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 31 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.455 - 0.571i)T + (-0.222 + 0.974i)T^{2} \) |
| 41 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 43 | \( 1 + (-0.142 - 0.0440i)T + (0.826 + 0.563i)T^{2} \) |
| 47 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 53 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 59 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 61 | \( 1 + (0.722 - 1.84i)T + (-0.733 - 0.680i)T^{2} \) |
| 67 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 73 | \( 1 + (0.147 + 1.97i)T + (-0.988 + 0.149i)T^{2} \) |
| 79 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 97 | \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.272275036329453988949938839993, −7.974873392659949248995732407354, −7.44697458297915841715242844115, −6.74326750978823895520122576165, −5.95435347842099572861028691308, −5.34755063367910896596980496289, −4.36803529511741887456010051610, −3.03610821136406886016652211852, −1.77129206736432517389912368752, −0.810027333582220085584566610188,
1.85737225359705134173490029618, 2.98375328098726880627085240391, 3.92144357190079124189058153225, 4.76057110007912049511957137611, 5.78577118237507408533616712545, 6.42223534450780712908314594848, 7.13937266305249065957400398680, 8.287876399959502536896262315269, 8.874007122399121893833677958188, 9.610342977035729824047619019629