L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·9-s + 4·11-s − 3·13-s − 14-s + 16-s − 17-s − 3·18-s + 4·22-s − 23-s − 3·26-s − 28-s − 4·31-s + 32-s − 34-s − 3·36-s + 11·37-s − 10·41-s − 2·43-s + 4·44-s − 46-s + 11·47-s + 49-s − 3·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s + 1.20·11-s − 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.707·18-s + 0.852·22-s − 0.208·23-s − 0.588·26-s − 0.188·28-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 1/2·36-s + 1.80·37-s − 1.56·41-s − 0.304·43-s + 0.603·44-s − 0.147·46-s + 1.60·47-s + 1/7·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.30614917040266229129834916137, −6.68940790572986483067064381183, −6.01202005800395902235385650023, −5.49724453957662851785330345356, −4.56679797372320399795364563211, −3.97031564998405657563814832929, −3.10310532824754280802888377210, −2.48195592761384522694417143693, −1.41998370484768490902893077241, 0,
1.41998370484768490902893077241, 2.48195592761384522694417143693, 3.10310532824754280802888377210, 3.97031564998405657563814832929, 4.56679797372320399795364563211, 5.49724453957662851785330345356, 6.01202005800395902235385650023, 6.68940790572986483067064381183, 7.30614917040266229129834916137