| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s − 11-s − 12-s + 13-s + 4·14-s + 16-s − 17-s + 18-s − 4·21-s − 22-s − 8·23-s − 24-s + 26-s − 27-s + 4·28-s − 2·29-s + 32-s + 33-s − 34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.872·21-s − 0.213·22-s − 1.66·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s + 0.176·32-s + 0.174·33-s − 0.171·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.66231124667715, −12.20492093022086, −11.79759152457977, −11.50033510818323, −10.96751586797678, −10.74269410519262, −10.15024231580924, −9.775740186419191, −8.993014027409579, −8.585956172850010, −7.990420640676976, −7.643045316018296, −7.322807289042478, −6.557006022493489, −6.114043784158724, −5.689402144941900, −5.260529944656406, −4.731181452179514, −4.329210761995083, −3.919063354354957, −3.315481040490543, −2.395244604075726, −2.127832203636747, −1.487722492476538, −0.9028858850773228, 0,
0.9028858850773228, 1.487722492476538, 2.127832203636747, 2.395244604075726, 3.315481040490543, 3.919063354354957, 4.329210761995083, 4.731181452179514, 5.260529944656406, 5.689402144941900, 6.114043784158724, 6.557006022493489, 7.322807289042478, 7.643045316018296, 7.990420640676976, 8.585956172850010, 8.993014027409579, 9.775740186419191, 10.15024231580924, 10.74269410519262, 10.96751586797678, 11.50033510818323, 11.79759152457977, 12.20492093022086, 12.66231124667715
