| L(s) = 1 | + 6·3-s − 94·5-s + 244·7-s − 207·9-s − 358·11-s + 770·13-s − 564·15-s + 670·17-s + 1.03e3·19-s + 1.46e3·21-s + 2.82e3·23-s + 5.71e3·25-s − 2.70e3·27-s + 762·29-s + 4.99e3·31-s − 2.14e3·33-s − 2.29e4·35-s + 3.56e3·37-s + 4.62e3·39-s + 858·41-s + 1.27e4·43-s + 1.94e4·45-s + 3.56e3·47-s + 4.27e4·49-s + 4.02e3·51-s + 9.11e3·53-s + 3.36e4·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 1.68·5-s + 1.88·7-s − 0.851·9-s − 0.892·11-s + 1.26·13-s − 0.647·15-s + 0.562·17-s + 0.654·19-s + 0.724·21-s + 1.11·23-s + 1.82·25-s − 0.712·27-s + 0.168·29-s + 0.932·31-s − 0.343·33-s − 3.16·35-s + 0.427·37-s + 0.486·39-s + 0.0797·41-s + 1.05·43-s + 1.43·45-s + 0.235·47-s + 2.54·49-s + 0.216·51-s + 0.445·53-s + 1.50·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.865597039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.865597039\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 94 T + p^{5} T^{2} \) |
| 7 | \( 1 - 244 T + p^{5} T^{2} \) |
| 11 | \( 1 + 358 T + p^{5} T^{2} \) |
| 13 | \( 1 - 770 T + p^{5} T^{2} \) |
| 17 | \( 1 - 670 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1030 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2828 T + p^{5} T^{2} \) |
| 29 | \( 1 - 762 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4992 T + p^{5} T^{2} \) |
| 37 | \( 1 - 3562 T + p^{5} T^{2} \) |
| 41 | \( 1 - 858 T + p^{5} T^{2} \) |
| 43 | \( 1 - 12786 T + p^{5} T^{2} \) |
| 47 | \( 1 - 3560 T + p^{5} T^{2} \) |
| 53 | \( 1 - 9114 T + p^{5} T^{2} \) |
| 59 | \( 1 + 8246 T + p^{5} T^{2} \) |
| 61 | \( 1 + 4414 T + p^{5} T^{2} \) |
| 67 | \( 1 + 29986 T + p^{5} T^{2} \) |
| 71 | \( 1 - 49572 T + p^{5} T^{2} \) |
| 73 | \( 1 + 24370 T + p^{5} T^{2} \) |
| 79 | \( 1 + 65176 T + p^{5} T^{2} \) |
| 83 | \( 1 + 39378 T + p^{5} T^{2} \) |
| 89 | \( 1 - 11134 T + p^{5} T^{2} \) |
| 97 | \( 1 - 478 T + p^{5} T^{2} \) |
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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
−12.10983436146925818710982555932, −11.30392660562263991452760106213, −10.84943262578513837876501770395, −8.720153423833104745471492567937, −8.119930571898606983640137097993, −7.50244198365188793256470642346, −5.42779102530448420779332860652, −4.29182005267925871549187187102, −2.97522705317463633019979943581, −0.962753042356626327274297015485,
0.962753042356626327274297015485, 2.97522705317463633019979943581, 4.29182005267925871549187187102, 5.42779102530448420779332860652, 7.50244198365188793256470642346, 8.119930571898606983640137097993, 8.720153423833104745471492567937, 10.84943262578513837876501770395, 11.30392660562263991452760106213, 12.10983436146925818710982555932
