| L(s) = 1 | + 2·2-s − 2·3-s + 4·4-s − 4·6-s + 7·7-s + 8·8-s − 23·9-s − 27·11-s − 8·12-s − 64·13-s + 14·14-s + 16·16-s − 24·17-s − 46·18-s + 62·19-s − 14·21-s − 54·22-s − 105·23-s − 16·24-s − 128·26-s + 100·27-s + 28·28-s + 141·29-s − 124·31-s + 32·32-s + 54·33-s − 48·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.384·3-s + 1/2·4-s − 0.272·6-s + 0.377·7-s + 0.353·8-s − 0.851·9-s − 0.740·11-s − 0.192·12-s − 1.36·13-s + 0.267·14-s + 1/4·16-s − 0.342·17-s − 0.602·18-s + 0.748·19-s − 0.145·21-s − 0.523·22-s − 0.951·23-s − 0.136·24-s − 0.965·26-s + 0.712·27-s + 0.188·28-s + 0.902·29-s − 0.718·31-s + 0.176·32-s + 0.284·33-s − 0.242·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 27 T + p^{3} T^{2} \) |
| 13 | \( 1 + 64 T + p^{3} T^{2} \) |
| 17 | \( 1 + 24 T + p^{3} T^{2} \) |
| 19 | \( 1 - 62 T + p^{3} T^{2} \) |
| 23 | \( 1 + 105 T + p^{3} T^{2} \) |
| 29 | \( 1 - 141 T + p^{3} T^{2} \) |
| 31 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 439 T + p^{3} T^{2} \) |
| 41 | \( 1 + 354 T + p^{3} T^{2} \) |
| 43 | \( 1 + 211 T + p^{3} T^{2} \) |
| 47 | \( 1 + 102 T + p^{3} T^{2} \) |
| 53 | \( 1 + 306 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 - 410 T + p^{3} T^{2} \) |
| 67 | \( 1 + 349 T + p^{3} T^{2} \) |
| 71 | \( 1 + 339 T + p^{3} T^{2} \) |
| 73 | \( 1 + 70 T + p^{3} T^{2} \) |
| 79 | \( 1 - 731 T + p^{3} T^{2} \) |
| 83 | \( 1 - 528 T + p^{3} T^{2} \) |
| 89 | \( 1 - 960 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1340 T + p^{3} T^{2} \) |
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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
−10.72367931748491725863492697064, −9.957465382211928028194531353597, −8.581241143215200589896359862162, −7.61430756385968017626171761744, −6.59743794219498824053514858249, −5.35615931614493167224466805079, −4.89727512665841417714669115349, −3.30556592802394084313675880243, −2.11853378601643820747991158462, 0,
2.11853378601643820747991158462, 3.30556592802394084313675880243, 4.89727512665841417714669115349, 5.35615931614493167224466805079, 6.59743794219498824053514858249, 7.61430756385968017626171761744, 8.581241143215200589896359862162, 9.957465382211928028194531353597, 10.72367931748491725863492697064
