| L(s) = 1 | + (−0.0871 + 0.996i)2-s + (1.26 − 1.18i)3-s + (−0.984 − 0.173i)4-s + (1.84 − 1.26i)5-s + (1.06 + 1.36i)6-s + (2.25 − 0.602i)7-s + (0.258 − 0.965i)8-s + (0.212 − 2.99i)9-s + (1.10 + 1.94i)10-s + (−2.69 + 1.55i)11-s + (−1.45 + 0.942i)12-s + (1.15 − 2.48i)13-s + (0.404 + 2.29i)14-s + (0.841 − 3.78i)15-s + (0.939 + 0.342i)16-s + (0.243 − 2.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.0616 + 0.704i)2-s + (0.731 − 0.681i)3-s + (−0.492 − 0.0868i)4-s + (0.824 − 0.566i)5-s + (0.435 + 0.557i)6-s + (0.850 − 0.227i)7-s + (0.0915 − 0.341i)8-s + (0.0706 − 0.997i)9-s + (0.347 + 0.615i)10-s + (−0.812 + 0.468i)11-s + (−0.419 + 0.272i)12-s + (0.321 − 0.689i)13-s + (0.108 + 0.613i)14-s + (0.217 − 0.976i)15-s + (0.234 + 0.0855i)16-s + (0.0591 − 0.676i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.97791 - 0.397782i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.97791 - 0.397782i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0871 - 0.996i)T \) |
| 3 | \( 1 + (-1.26 + 1.18i)T \) |
| 5 | \( 1 + (-1.84 + 1.26i)T \) |
| 19 | \( 1 + (1.91 - 3.91i)T \) |
| good | 7 | \( 1 + (-2.25 + 0.602i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (2.69 - 1.55i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.15 + 2.48i)T + (-8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.243 + 2.78i)T + (-16.7 - 2.95i)T^{2} \) |
| 23 | \( 1 + (4.18 - 2.92i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (-2.63 - 2.20i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.72 + 2.97i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.700 + 0.700i)T + 37iT^{2} \) |
| 41 | \( 1 + (2.48 - 6.81i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-7.16 - 5.01i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-10.1 + 0.884i)T + (46.2 - 8.16i)T^{2} \) |
| 53 | \( 1 + (-2.75 + 1.92i)T + (18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (-7.25 + 6.08i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.249 + 1.41i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.34 - 15.3i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (4.11 - 0.725i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (12.9 - 6.03i)T + (46.9 - 55.9i)T^{2} \) |
| 79 | \( 1 + (3.12 - 8.58i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.496 + 0.132i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-5.50 + 2.00i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (5.20 + 0.455i)T + (95.5 + 16.8i)T^{2} \) |
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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
−10.31903020062335686410219655647, −9.683947134819029686410280892661, −8.597140514859504445588215677254, −8.032882378613471607497923618264, −7.32895387337119514371292924347, −6.11494318997878008240410232023, −5.32370483555244050370272336911, −4.18058547297968920699902277402, −2.52148375063300273470298337356, −1.24170060430962277955765358940,
1.96244027553719265249460718265, 2.69964610104156799314119302846, 3.99154319869360198229955750570, 4.98381799789463593235146964822, 6.02489174383948107141327655711, 7.47134433045917342728068342231, 8.591961989782872425203235882213, 8.955862891001562192577241456591, 10.22740313014168667620022020827, 10.55900316307616206492771527390
