| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 25·5-s − 36·6-s − 188·7-s + 64·8-s + 81·9-s + 100·10-s + 135·11-s − 144·12-s − 1.11e3·13-s − 752·14-s − 225·15-s + 256·16-s + 289·17-s + 324·18-s − 929·19-s + 400·20-s + 1.69e3·21-s + 540·22-s − 1.70e3·23-s − 576·24-s − 2.50e3·25-s − 4.45e3·26-s − 729·27-s − 3.00e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.45·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.336·11-s − 0.288·12-s − 1.82·13-s − 1.02·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.590·19-s + 0.223·20-s + 0.837·21-s + 0.237·22-s − 0.670·23-s − 0.204·24-s − 4/5·25-s − 1.29·26-s − 0.192·27-s − 0.725·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 17 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - p^{2} T + p^{5} T^{2} \) |
| 7 | \( 1 + 188 T + p^{5} T^{2} \) |
| 11 | \( 1 - 135 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1113 T + p^{5} T^{2} \) |
| 19 | \( 1 + 929 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1701 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4670 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5490 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10808 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8911 T + p^{5} T^{2} \) |
| 43 | \( 1 + 11839 T + p^{5} T^{2} \) |
| 47 | \( 1 + 8578 T + p^{5} T^{2} \) |
| 53 | \( 1 - 35606 T + p^{5} T^{2} \) |
| 59 | \( 1 - 6138 T + p^{5} T^{2} \) |
| 61 | \( 1 - 19688 T + p^{5} T^{2} \) |
| 67 | \( 1 + 14724 T + p^{5} T^{2} \) |
| 71 | \( 1 - 48396 T + p^{5} T^{2} \) |
| 73 | \( 1 - 10322 T + p^{5} T^{2} \) |
| 79 | \( 1 - 77710 T + p^{5} T^{2} \) |
| 83 | \( 1 + 108258 T + p^{5} T^{2} \) |
| 89 | \( 1 + 116488 T + p^{5} T^{2} \) |
| 97 | \( 1 - 10524 T + p^{5} T^{2} \) |
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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
−12.55569026920709517747529036156, −11.56260070584639553201201968936, −10.11017068855536688590645278183, −9.518582601109329034051943982100, −7.41177845963238988847581320770, −6.38865244873897904836318394937, −5.42737028916906673174536848644, −3.92035120172224672601482816080, −2.33020120270923248137489389031, 0,
2.33020120270923248137489389031, 3.92035120172224672601482816080, 5.42737028916906673174536848644, 6.38865244873897904836318394937, 7.41177845963238988847581320770, 9.518582601109329034051943982100, 10.11017068855536688590645278183, 11.56260070584639553201201968936, 12.55569026920709517747529036156
