L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 5·11-s + 12-s − 3·13-s − 14-s + 16-s − 2·17-s + 18-s − 3·19-s − 21-s − 5·22-s − 23-s + 24-s − 3·26-s + 27-s − 28-s − 29-s + 32-s − 5·33-s − 2·34-s + 36-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 − 1.50·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.688·19-s − 0.218·21-s − 1.06·22-s − 0.208·23-s + 0.204·24-s − 0.588·26-s + 0.192·27-s − 0.188·28-s − 0.185·29-s + 0.176·32-s − 0.870·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.139305170924357060414030993493, −7.41876380247308632126018982395, −6.79187132270282611002857311265, −5.86846073750613392751308448746, −5.05274234599188544061156257914, −4.41344662560399238752812311537, −3.38598205999351720885912206458, −2.64998765900272935409891220058, −1.93106433091937374023502636476, 0,
1.93106433091937374023502636476, 2.64998765900272935409891220058, 3.38598205999351720885912206458, 4.41344662560399238752812311537, 5.05274234599188544061156257914, 5.86846073750613392751308448746, 6.79187132270282611002857311265, 7.41876380247308632126018982395, 8.139305170924357060414030993493