| L(s) = 1 | + 2·7-s + 11-s + 13-s − 6·17-s + 2·19-s − 5·25-s − 6·29-s − 10·31-s + 2·37-s + 8·43-s − 12·47-s − 3·49-s − 6·53-s + 2·61-s + 2·67-s − 10·73-s + 2·77-s − 4·79-s + 2·91-s + 14·97-s − 6·101-s + 8·103-s + 12·107-s − 10·109-s − 6·113-s − 12·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s + 0.277·13-s − 1.45·17-s + 0.458·19-s − 25-s − 1.11·29-s − 1.79·31-s + 0.328·37-s + 1.21·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.256·61-s + 0.244·67-s − 1.17·73-s + 0.227·77-s − 0.450·79-s + 0.209·91-s + 1.42·97-s − 0.597·101-s + 0.788·103-s + 1.16·107-s − 0.957·109-s − 0.564·113-s − 1.10·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77834757777646485461776861295, −7.27989797211997650139080607488, −6.38181048674495160130889708917, −5.69598757160348880490297289545, −4.89036566000888705421230357053, −4.14878402511126597991215328979, −3.40977723585453614519207326902, −2.18110332635051037682013804804, −1.53456290706493550194033500810, 0,
1.53456290706493550194033500810, 2.18110332635051037682013804804, 3.40977723585453614519207326902, 4.14878402511126597991215328979, 4.89036566000888705421230357053, 5.69598757160348880490297289545, 6.38181048674495160130889708917, 7.27989797211997650139080607488, 7.77834757777646485461776861295
