L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s − 2·13-s + 16-s − 2·17-s + 18-s − 2·19-s − 8·23-s + 24-s − 2·26-s + 27-s − 8·29-s − 4·31-s + 32-s − 2·34-s + 36-s + 6·37-s − 2·38-s − 2·39-s − 10·41-s − 2·43-s − 8·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.458·19-s − 1.66·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s − 1.48·29-s − 0.718·31-s + 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.986·37-s − 0.324·38-s − 0.320·39-s − 1.56·41-s − 0.304·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.75972303732766610653670292427, −6.71746052302026106008361856652, −6.23649943255438050233522172382, −5.33714204239010871677222163923, −4.64680835550842315040701992404, −3.88284043167385451077899255203, −3.29818149609701544272153132242, −2.23452117332795942922270569875, −1.76423348275487500936740982432, 0,
1.76423348275487500936740982432, 2.23452117332795942922270569875, 3.29818149609701544272153132242, 3.88284043167385451077899255203, 4.64680835550842315040701992404, 5.33714204239010871677222163923, 6.23649943255438050233522172382, 6.71746052302026106008361856652, 7.75972303732766610653670292427