| L(s) = 1 | + 2-s + 4-s + 5-s − 2·7-s + 8-s + 10-s − 5·11-s + 13-s − 2·14-s + 16-s − 3·17-s + 20-s − 5·22-s + 4·23-s + 25-s + 26-s − 2·28-s + 29-s − 5·31-s + 32-s − 3·34-s − 2·35-s + 6·37-s + 40-s + 9·43-s − 5·44-s + 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s − 1.50·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.223·20-s − 1.06·22-s + 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s + 0.185·29-s − 0.898·31-s + 0.176·32-s − 0.514·34-s − 0.338·35-s + 0.986·37-s + 0.158·40-s + 1.37·43-s − 0.753·44-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.32502651302536, −14.71587481348754, −14.04499788319974, −13.63514552640880, −13.02343153482763, −12.80778202399444, −12.43301688314831, −11.39147349418583, −11.13140488281965, −10.39107392529352, −10.19605295334211, −9.238608814263679, −8.982775329842323, −8.122700304206053, −7.451324086609883, −7.069538301394268, −6.272506628382776, −5.835491870557695, −5.321575909867656, −4.635445078978172, −4.044566402186832, −3.181347199068308, −2.671932566668155, −2.168965106957093, −1.070283428290364, 0,
1.070283428290364, 2.168965106957093, 2.671932566668155, 3.181347199068308, 4.044566402186832, 4.635445078978172, 5.321575909867656, 5.835491870557695, 6.272506628382776, 7.069538301394268, 7.451324086609883, 8.122700304206053, 8.982775329842323, 9.238608814263679, 10.19605295334211, 10.39107392529352, 11.13140488281965, 11.39147349418583, 12.43301688314831, 12.80778202399444, 13.02343153482763, 13.63514552640880, 14.04499788319974, 14.71587481348754, 15.32502651302536
