| L(s) = 1 | + 16·2-s + 81·3-s + 256·4-s + 1.29e3·6-s + 7.16e3·7-s + 4.09e3·8-s + 6.56e3·9-s − 8.37e4·11-s + 2.07e4·12-s − 1.28e5·13-s + 1.14e5·14-s + 6.55e4·16-s − 5.60e5·17-s + 1.04e5·18-s − 5.77e5·19-s + 5.80e5·21-s − 1.33e6·22-s − 2.43e6·23-s + 3.31e5·24-s − 2.05e6·26-s + 5.31e5·27-s + 1.83e6·28-s + 5.79e6·29-s + 4.14e6·31-s + 1.04e6·32-s − 6.78e6·33-s − 8.97e6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.12·7-s + 0.353·8-s + 1/3·9-s − 1.72·11-s + 0.288·12-s − 1.24·13-s + 0.797·14-s + 1/4·16-s − 1.62·17-s + 0.235·18-s − 1.01·19-s + 0.651·21-s − 1.21·22-s − 1.81·23-s + 0.204·24-s − 0.879·26-s + 0.192·27-s + 0.564·28-s + 1.52·29-s + 0.806·31-s + 0.176·32-s − 0.995·33-s − 1.15·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 3 | \( 1 - p^{4} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1024 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 83748 T + p^{9} T^{2} \) |
| 13 | \( 1 + 128126 T + p^{9} T^{2} \) |
| 17 | \( 1 + 560802 T + p^{9} T^{2} \) |
| 19 | \( 1 + 577660 T + p^{9} T^{2} \) |
| 23 | \( 1 + 2431296 T + p^{9} T^{2} \) |
| 29 | \( 1 - 5791710 T + p^{9} T^{2} \) |
| 31 | \( 1 - 4145312 T + p^{9} T^{2} \) |
| 37 | \( 1 - 7011658 T + p^{9} T^{2} \) |
| 41 | \( 1 + 8881398 T + p^{9} T^{2} \) |
| 43 | \( 1 - 15730684 T + p^{9} T^{2} \) |
| 47 | \( 1 + 60552072 T + p^{9} T^{2} \) |
| 53 | \( 1 + 30273366 T + p^{9} T^{2} \) |
| 59 | \( 1 - 45957660 T + p^{9} T^{2} \) |
| 61 | \( 1 - 37595102 T + p^{9} T^{2} \) |
| 67 | \( 1 + 196784012 T + p^{9} T^{2} \) |
| 71 | \( 1 - 56047992 T + p^{9} T^{2} \) |
| 73 | \( 1 - 159688054 T + p^{9} T^{2} \) |
| 79 | \( 1 - 201923360 T + p^{9} T^{2} \) |
| 83 | \( 1 - 362955444 T + p^{9} T^{2} \) |
| 89 | \( 1 + 272479110 T + p^{9} T^{2} \) |
| 97 | \( 1 - 600852478 T + p^{9} T^{2} \) |
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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
−10.82406873119820077049203247711, −10.00076759383578778619451400780, −8.331569960468808725671640575694, −7.82392286794871820817538321191, −6.47684746132687521916550586520, −4.95368146715081362388834695612, −4.40970475776097495911730335598, −2.60004252948876051545657198247, −2.04741435182661667525953203407, 0,
2.04741435182661667525953203407, 2.60004252948876051545657198247, 4.40970475776097495911730335598, 4.95368146715081362388834695612, 6.47684746132687521916550586520, 7.82392286794871820817538321191, 8.331569960468808725671640575694, 10.00076759383578778619451400780, 10.82406873119820077049203247711
