| L(s) = 1 | − 13.3i·2-s + 163. i·3-s + 333.·4-s + 2.18e3·6-s − 2.40e3i·7-s − 1.12e4i·8-s − 7.02e3·9-s − 9.01e4·11-s + 5.44e4i·12-s − 3.19e3i·13-s − 3.20e4·14-s + 1.98e4·16-s − 1.16e5i·17-s + 9.38e4i·18-s + 1.42e5·19-s + ⋯ |
| L(s) = 1 | − 0.590i·2-s + 1.16i·3-s + 0.651·4-s + 0.687·6-s − 0.377i·7-s − 0.975i·8-s − 0.356·9-s − 1.85·11-s + 0.758i·12-s − 0.0310i·13-s − 0.223·14-s + 0.0756·16-s − 0.338i·17-s + 0.210i·18-s + 0.250·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.429304406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.429304406\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 2.40e3iT \) |
| good | 2 | \( 1 + 13.3iT - 512T^{2} \) |
| 3 | \( 1 - 163. iT - 1.96e4T^{2} \) |
| 11 | \( 1 + 9.01e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.19e3iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 1.16e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 - 1.42e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.27e6iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 1.42e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.67e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 1.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.97e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 1.07e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 - 7.07e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 6.40e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.69e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.16e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 1.44e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.60e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 4.89e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.31e7iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 2.08e6T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.15e8iT - 7.60e17T^{2} \) |
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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
−10.83528286333223257407560110660, −10.26272065347909706325050488133, −9.629325723665753692087251597417, −8.081553816165641229269226757972, −7.06784005264015917951154514935, −5.57124034942870358947595683269, −4.53264872413193179460655917773, −3.33483757365466571059116675975, −2.46103927508604375301394337756, −0.866351364759143065827533756803,
0.65959112185435319163336968953, 2.10562849788965077745367250442, 2.75518979994858561130237228565, 4.94255316455507408931999102656, 6.03324718527696366176242801086, 6.82364418855635213428000628451, 7.889462811636004212312028419465, 8.256849424830678280737041346218, 10.08259875498127774529876549206, 11.05919961201444245693184129245
