| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 2·11-s − 6·13-s + 14-s + 16-s + 2·17-s + 6·19-s − 2·22-s − 23-s + 6·26-s − 28-s − 9·29-s − 2·31-s − 32-s − 2·34-s + 2·37-s − 6·38-s + 11·41-s − 4·43-s + 2·44-s + 46-s + 7·47-s − 6·49-s − 6·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.603·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 1.37·19-s − 0.426·22-s − 0.208·23-s + 1.17·26-s − 0.188·28-s − 1.67·29-s − 0.359·31-s − 0.176·32-s − 0.342·34-s + 0.328·37-s − 0.973·38-s + 1.71·41-s − 0.609·43-s + 0.301·44-s + 0.147·46-s + 1.02·47-s − 6/7·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.83363987662291462133290912027, −7.51054612699914631844435303243, −6.85375087117219460892529582818, −5.86969935881180331945520030129, −5.25833616399397912417664210182, −4.18242956185978364798445194529, −3.23108286709011396574644230445, −2.39873837184369674455978706927, −1.29732355983093497181986455091, 0,
1.29732355983093497181986455091, 2.39873837184369674455978706927, 3.23108286709011396574644230445, 4.18242956185978364798445194529, 5.25833616399397912417664210182, 5.86969935881180331945520030129, 6.85375087117219460892529582818, 7.51054612699914631844435303243, 7.83363987662291462133290912027
