L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 3·11-s − 4·13-s − 14-s + 16-s − 3·17-s + 2·19-s + 20-s − 3·22-s − 6·23-s + 25-s − 4·26-s − 28-s − 3·29-s + 5·31-s + 32-s − 3·34-s − 35-s + 37-s + 2·38-s + 40-s − 3·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.904·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s + 0.223·20-s − 0.639·22-s − 1.25·23-s + 1/5·25-s − 0.784·26-s − 0.188·28-s − 0.557·29-s + 0.898·31-s + 0.176·32-s − 0.514·34-s − 0.169·35-s + 0.164·37-s + 0.324·38-s + 0.158·40-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.037658727484305993130701016237, −7.48472813821623192367156751152, −6.56330850429489211825831149583, −5.99940287722666102586658741510, −5.06620453918957560960032027985, −4.59767259681282669603710565857, −3.42767809661089504801834268834, −2.62659393830675313979656999849, −1.82989205635176673644713429623, 0,
1.82989205635176673644713429623, 2.62659393830675313979656999849, 3.42767809661089504801834268834, 4.59767259681282669603710565857, 5.06620453918957560960032027985, 5.99940287722666102586658741510, 6.56330850429489211825831149583, 7.48472813821623192367156751152, 8.037658727484305993130701016237