| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 3·11-s − 12·12-s − 4·13-s + 14·14-s + 16·16-s − 54·17-s + 18·18-s − 148·19-s − 21·21-s + 6·22-s − 15·23-s − 24·24-s − 8·26-s − 27·27-s + 28·28-s − 69·29-s + 146·31-s + 32·32-s − 9·33-s − 108·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.0822·11-s − 0.288·12-s − 0.0853·13-s + 0.267·14-s + 1/4·16-s − 0.770·17-s + 0.235·18-s − 1.78·19-s − 0.218·21-s + 0.0581·22-s − 0.135·23-s − 0.204·24-s − 0.0603·26-s − 0.192·27-s + 0.188·28-s − 0.441·29-s + 0.845·31-s + 0.176·32-s − 0.0474·33-s − 0.544·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 - 3 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 148 T + p^{3} T^{2} \) |
| 23 | \( 1 + 15 T + p^{3} T^{2} \) |
| 29 | \( 1 + 69 T + p^{3} T^{2} \) |
| 31 | \( 1 - 146 T + p^{3} T^{2} \) |
| 37 | \( 1 + 19 T + p^{3} T^{2} \) |
| 41 | \( 1 + 24 T + p^{3} T^{2} \) |
| 43 | \( 1 - 29 T + p^{3} T^{2} \) |
| 47 | \( 1 - 228 T + p^{3} T^{2} \) |
| 53 | \( 1 - 174 T + p^{3} T^{2} \) |
| 59 | \( 1 + 732 T + p^{3} T^{2} \) |
| 61 | \( 1 + 220 T + p^{3} T^{2} \) |
| 67 | \( 1 - 11 T + p^{3} T^{2} \) |
| 71 | \( 1 + 429 T + p^{3} T^{2} \) |
| 73 | \( 1 + 910 T + p^{3} T^{2} \) |
| 79 | \( 1 + 889 T + p^{3} T^{2} \) |
| 83 | \( 1 - 78 T + p^{3} T^{2} \) |
| 89 | \( 1 + 960 T + p^{3} T^{2} \) |
| 97 | \( 1 + 550 T + p^{3} T^{2} \) |
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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
−9.080092444301629143177282181403, −8.225824395405727375514973435901, −7.19975290076108623472897281059, −6.40656102382541542719325650466, −5.70489927379339248664058328170, −4.60505674122055693810162266028, −4.11843806398697833885320265686, −2.66200470265353487074124269293, −1.59467517800723947711890097254, 0,
1.59467517800723947711890097254, 2.66200470265353487074124269293, 4.11843806398697833885320265686, 4.60505674122055693810162266028, 5.70489927379339248664058328170, 6.40656102382541542719325650466, 7.19975290076108623472897281059, 8.225824395405727375514973435901, 9.080092444301629143177282181403
