| L(s) = 1 | − 18·5-s − 36·11-s + 10·13-s + 18·17-s + 100·19-s − 72·23-s + 199·25-s + 234·29-s + 16·31-s − 226·37-s + 90·41-s + 452·43-s + 432·47-s − 414·53-s + 648·55-s − 684·59-s − 422·61-s − 180·65-s + 332·67-s + 360·71-s − 26·73-s + 512·79-s − 1.18e3·83-s − 324·85-s − 630·89-s − 1.80e3·95-s + 1.05e3·97-s + ⋯ |
| L(s) = 1 | − 1.60·5-s − 0.986·11-s + 0.213·13-s + 0.256·17-s + 1.20·19-s − 0.652·23-s + 1.59·25-s + 1.49·29-s + 0.0926·31-s − 1.00·37-s + 0.342·41-s + 1.60·43-s + 1.34·47-s − 1.07·53-s + 1.58·55-s − 1.50·59-s − 0.885·61-s − 0.343·65-s + 0.605·67-s + 0.601·71-s − 0.0416·73-s + 0.729·79-s − 1.57·83-s − 0.413·85-s − 0.750·89-s − 1.94·95-s + 1.10·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 234 T + p^{3} T^{2} \) |
| 31 | \( 1 - 16 T + p^{3} T^{2} \) |
| 37 | \( 1 + 226 T + p^{3} T^{2} \) |
| 41 | \( 1 - 90 T + p^{3} T^{2} \) |
| 43 | \( 1 - 452 T + p^{3} T^{2} \) |
| 47 | \( 1 - 432 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 684 T + p^{3} T^{2} \) |
| 61 | \( 1 + 422 T + p^{3} T^{2} \) |
| 67 | \( 1 - 332 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 26 T + p^{3} T^{2} \) |
| 79 | \( 1 - 512 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 630 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1054 T + p^{3} T^{2} \) |
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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
−8.337617908676463810725599056245, −7.72277839900436686486903202696, −7.28651124892840429238187796598, −6.13749555685334373095846121630, −5.12853804731705798215557659125, −4.34692342390994105346955289328, −3.46010965783694822408001880981, −2.67721558033884666827525332055, −1.03091077808027250966867992298, 0,
1.03091077808027250966867992298, 2.67721558033884666827525332055, 3.46010965783694822408001880981, 4.34692342390994105346955289328, 5.12853804731705798215557659125, 6.13749555685334373095846121630, 7.28651124892840429238187796598, 7.72277839900436686486903202696, 8.337617908676463810725599056245
