| L(s) = 1 | + (−27.4 + 15.8i)5-s + (37.6 − 65.2i)7-s + (123. + 71.1i)11-s + (96.3 + 166. i)13-s + 325. i·17-s + 314.·19-s + (443. − 256. i)23-s + (188. − 326. i)25-s + (136. + 78.8i)29-s + (183. + 318. i)31-s + 2.38e3i·35-s + 1.73e3·37-s + (342. − 197. i)41-s + (−360. + 624. i)43-s + (−2.15e3 − 1.24e3i)47-s + ⋯ |
| L(s) = 1 | + (−1.09 + 0.633i)5-s + (0.769 − 1.33i)7-s + (1.01 + 0.588i)11-s + (0.569 + 0.987i)13-s + 1.12i·17-s + 0.870·19-s + (0.839 − 0.484i)23-s + (0.301 − 0.522i)25-s + (0.162 + 0.0937i)29-s + (0.191 + 0.331i)31-s + 1.94i·35-s + 1.26·37-s + (0.203 − 0.117i)41-s + (−0.194 + 0.337i)43-s + (−0.974 − 0.562i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.59213 + 0.408660i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59213 + 0.408660i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (27.4 - 15.8i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (-37.6 + 65.2i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-123. - 71.1i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-96.3 - 166. i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 - 325. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 314.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-443. + 256. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-136. - 78.8i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-183. - 318. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.73e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-342. + 197. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (360. - 624. i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (2.15e3 + 1.24e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + 3.98e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-2.13e3 + 1.23e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.24e3 - 2.14e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.29e3 - 5.71e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 5.82e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.79e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-1.93e3 + 3.35e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (1.04e4 + 6.01e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 - 7.63e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.45e3 + 2.52e3i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.12589645559710564462067820068, −11.64179141015985188637800661001, −11.21602449316188236608783509625, −10.06148279589825324926246049246, −8.535448453231649719435685742854, −7.40781396727476557778019928321, −6.66468366119843418075814248433, −4.45089109264747375366187411470, −3.69375349289803353284616410463, −1.28004882657192918321811050719,
0.949928618562229170525563108687, 3.13900122944042677893112737325, 4.72265259848176459687293811234, 5.84777766530249678316749024603, 7.63806481642167766905892083409, 8.530513325829332297129357384938, 9.353872666618889105825153295056, 11.34515158248514137139066377267, 11.65815535474157293868852627574, 12.67623836926280670128742653794
