| L(s) = 1 | + (0.792 + 1.77i)2-s + (−1.79 − 1.61i)3-s + (−1.20 + 1.33i)4-s + (2.07 − 0.842i)5-s + (1.45 − 4.46i)6-s + (2.52 + 0.802i)7-s + (0.377 + 0.122i)8-s + (0.293 + 2.79i)9-s + (3.14 + 3.01i)10-s + (−0.259 + 2.46i)11-s + (4.30 − 0.452i)12-s + (−2.29 − 3.16i)13-s + (0.569 + 5.12i)14-s + (−5.06 − 1.83i)15-s + (0.456 + 4.34i)16-s + (−0.643 − 3.02i)17-s + ⋯ |
| L(s) = 1 | + (0.560 + 1.25i)2-s + (−1.03 − 0.930i)3-s + (−0.600 + 0.667i)4-s + (0.926 − 0.376i)5-s + (0.592 − 1.82i)6-s + (0.952 + 0.303i)7-s + (0.133 + 0.0433i)8-s + (0.0977 + 0.930i)9-s + (0.993 + 0.954i)10-s + (−0.0782 + 0.744i)11-s + (1.24 − 0.130i)12-s + (−0.637 − 0.876i)13-s + (0.152 + 1.36i)14-s + (−1.30 − 0.472i)15-s + (0.114 + 1.08i)16-s + (−0.156 − 0.734i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29583 + 0.484758i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29583 + 0.484758i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.07 + 0.842i)T \) |
| 7 | \( 1 + (-2.52 - 0.802i)T \) |
| good | 2 | \( 1 + (-0.792 - 1.77i)T + (-1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (1.79 + 1.61i)T + (0.313 + 2.98i)T^{2} \) |
| 11 | \( 1 + (0.259 - 2.46i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (2.29 + 3.16i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.643 + 3.02i)T + (-15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-3.42 - 3.80i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (1.85 + 4.16i)T + (-15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (0.797 + 2.45i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.17 - 1.31i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (8.11 - 0.852i)T + (36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (4.21 - 3.06i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 5.80iT - 43T^{2} \) |
| 47 | \( 1 + (0.453 - 2.13i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (5.71 + 5.14i)T + (5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (10.0 + 4.49i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-7.90 + 3.52i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (1.78 + 8.38i)T + (-61.2 + 27.2i)T^{2} \) |
| 71 | \( 1 + (-4.36 - 13.4i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-12.2 - 1.28i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-6.11 - 1.30i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (4.78 + 1.55i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (12.6 - 5.65i)T + (59.5 - 66.1i)T^{2} \) |
| 97 | \( 1 + (-4.30 + 1.39i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.79124741358263150834662093474, −12.29688052633274174630352809441, −11.05118381437097416229324057806, −9.816288088300861881247899522235, −8.199136895623892748837810856617, −7.33930661516494392745518920603, −6.37438191079846711013908819344, −5.36323380701871390629814005988, −4.96309997011422940846915679552, −1.77528851648036103962255638230,
1.86786591939205072969579926848, 3.64122571870641610840717235125, 4.87301528043416461054931466462, 5.57966242465741940576934057627, 7.18297283182448250289706694390, 9.150721448180164352102044579802, 10.14713694180775673746760452553, 10.88518730616746367993056282064, 11.33785311031162201358632917569, 12.23710041008195984442793486041
