| L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.413 + 0.459i)3-s + (0.669 + 0.743i)4-s + (1.11 + 1.93i)5-s + (−0.190 − 0.587i)6-s + (−0.5 − 2.59i)7-s + (−0.309 − 0.951i)8-s + (0.273 − 2.60i)9-s + (−0.233 − 2.22i)10-s + (−0.402 − 3.83i)11-s + (−0.0646 + 0.614i)12-s + (4.73 + 3.44i)13-s + (−0.599 + 2.57i)14-s + (−0.427 + 1.31i)15-s + (−0.104 + 0.994i)16-s + (7.53 + 1.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 − 0.287i)2-s + (0.238 + 0.265i)3-s + (0.334 + 0.371i)4-s + (0.499 + 0.866i)5-s + (−0.0779 − 0.239i)6-s + (−0.188 − 0.981i)7-s + (−0.109 − 0.336i)8-s + (0.0912 − 0.867i)9-s + (−0.0739 − 0.703i)10-s + (−0.121 − 1.15i)11-s + (−0.0186 + 0.177i)12-s + (1.31 + 0.954i)13-s + (−0.160 + 0.688i)14-s + (−0.110 + 0.339i)15-s + (−0.0261 + 0.248i)16-s + (1.82 + 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20865 - 0.100832i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20865 - 0.100832i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.913 + 0.406i)T \) |
| 5 | \( 1 + (-1.11 - 1.93i)T \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
| good | 3 | \( 1 + (-0.413 - 0.459i)T + (-0.313 + 2.98i)T^{2} \) |
| 11 | \( 1 + (0.402 + 3.83i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-4.73 - 3.44i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-7.53 - 1.60i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (4.01 - 4.45i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.34 - 0.598i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (-1.07 + 3.30i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.74 - 1.00i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (0.0399 - 0.379i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (7.59 + 5.51i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + (9.03 - 1.92i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (0.157 + 0.175i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (4.86 - 2.16i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-7.52 - 3.34i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-0.978 - 0.207i)T + (61.2 + 27.2i)T^{2} \) |
| 71 | \( 1 + (-4.92 + 15.1i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.562 + 5.35i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (8.74 - 1.85i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (1.78 + 5.48i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (5.39 + 2.40i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (3.47 - 10.6i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.17601839162740646284984597103, −10.37290871755347084753348198174, −9.863328875066671051344254516906, −8.748064933766401304042935834017, −7.87659539842052572480379012380, −6.57277554229539843334832726513, −6.05832059742523877731276782679, −3.79152194979194864491230283796, −3.30592872042852019200095054648, −1.31927912841159955819862572174,
1.45033001612606641119532428359, 2.76652289522969313347207641700, 4.90684022823063119485744868146, 5.63089415269874026717040162449, 6.81325066220333825660429742233, 8.153591777961632481100416735058, 8.469112926496337081358759948049, 9.632087958580456990789389045564, 10.26110347657593826416925769448, 11.47042936330091345066358046765
