| L(s) = 1 | + (3.80 + 1.23i)2-s + (−16.9 + 23.3i)3-s + (12.9 + 9.40i)4-s + (−8.25 + 55.2i)5-s + (−93.3 + 67.8i)6-s − 138. i·7-s + (37.6 + 51.7i)8-s + (−182. − 560. i)9-s + (−99.7 + 200. i)10-s + (−71.4 + 219. i)11-s + (−439. + 142. i)12-s + (−251. + 81.8i)13-s + (171. − 526. i)14-s + (−1.15e3 − 1.13e3i)15-s + (79.1 + 243. i)16-s + (323. + 445. i)17-s + ⋯ |
| L(s) = 1 | + (0.672 + 0.218i)2-s + (−1.08 + 1.49i)3-s + (0.404 + 0.293i)4-s + (−0.147 + 0.989i)5-s + (−1.05 + 0.769i)6-s − 1.06i·7-s + (0.207 + 0.286i)8-s + (−0.749 − 2.30i)9-s + (−0.315 + 0.632i)10-s + (−0.177 + 0.547i)11-s + (−0.880 + 0.285i)12-s + (−0.413 + 0.134i)13-s + (0.233 − 0.718i)14-s + (−1.32 − 1.29i)15-s + (0.0772 + 0.237i)16-s + (0.271 + 0.373i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.265i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.144838 - 1.06959i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144838 - 1.06959i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.80 - 1.23i)T \) |
| 5 | \( 1 + (8.25 - 55.2i)T \) |
| good | 3 | \( 1 + (16.9 - 23.3i)T + (-75.0 - 231. i)T^{2} \) |
| 7 | \( 1 + 138. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (71.4 - 219. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (251. - 81.8i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-323. - 445. i)T + (-4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (1.50e3 - 1.09e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-3.18e3 - 1.03e3i)T + (5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (3.65e3 + 2.65e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (8.60e3 - 6.25e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.21e4 + 3.93e3i)T + (5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-2.16e3 - 6.65e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.45e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (9.93e3 - 1.36e4i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-7.60e3 + 1.04e4i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-7.68e3 - 2.36e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-6.87e3 + 2.11e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-8.84e3 - 1.21e4i)T + (-4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (3.71e4 + 2.70e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (2.76e4 + 8.97e3i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.02e4 - 1.47e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-4.03e4 - 5.55e4i)T + (-1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-8.33e3 + 2.56e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (2.60e4 - 3.58e4i)T + (-2.65e9 - 8.16e9i)T^{2} \) |
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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
−14.93029877688605327588959039950, −14.64360710703142393140562095061, −12.78920171270631879453543787481, −11.30992859657311805734229395086, −10.70738922397339231387934383161, −9.736810090991600669541152539302, −7.30255817525397374270218717881, −6.03439524855819419557260803106, −4.58122565128455238209083123720, −3.56638974614200797463160775537,
0.49236548187930232415672240539, 2.17893053342935311906089823204, 5.10378333777631792972478016562, 5.88286594961452223918270614527, 7.33792183428714162178460440392, 8.817099884743075257819394778532, 11.08661754059947077667494003318, 11.88187481992807879424487254305, 12.83374090996967776650746035282, 13.21227461299628441481283937907
