L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 4·11-s + 4·13-s − 14-s + 16-s + 2·17-s − 2·19-s − 4·22-s + 23-s + 4·26-s − 28-s + 32-s + 2·34-s − 2·37-s − 2·38-s + 12·41-s + 4·43-s − 4·44-s + 46-s + 4·47-s + 49-s + 4·52-s − 14·53-s − 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.20·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.458·19-s − 0.852·22-s + 0.208·23-s + 0.784·26-s − 0.188·28-s + 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.324·38-s + 1.87·41-s + 0.609·43-s − 0.603·44-s + 0.147·46-s + 0.583·47-s + 1/7·49-s + 0.554·52-s − 1.92·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−14.39455863430703, −13.68419529791945, −13.41643387711854, −12.77406560662177, −12.62318636241765, −11.96431689997969, −11.31417146988030, −10.80610335086531, −10.54357573621294, −9.961901328834416, −9.222315761191586, −8.786548211998797, −8.130924667454907, −7.476975250013181, −7.325663746496101, −6.230998195147733, −6.058430870560726, −5.598003627749552, −4.724827018932592, −4.429524018883852, −3.608109792880197, −3.085679057235485, −2.603881577169787, −1.781013662188228, −1.013346790865327, 0,
1.013346790865327, 1.781013662188228, 2.603881577169787, 3.085679057235485, 3.608109792880197, 4.429524018883852, 4.724827018932592, 5.598003627749552, 6.058430870560726, 6.230998195147733, 7.325663746496101, 7.476975250013181, 8.130924667454907, 8.786548211998797, 9.222315761191586, 9.961901328834416, 10.54357573621294, 10.80610335086531, 11.31417146988030, 11.96431689997969, 12.62318636241765, 12.77406560662177, 13.41643387711854, 13.68419529791945, 14.39455863430703