L(s) = 1 | + 7-s − 4·11-s + 6·17-s − 4·19-s − 4·23-s − 5·25-s − 2·29-s + 8·31-s + 6·37-s + 6·41-s + 4·43-s + 6·47-s + 49-s − 6·53-s − 12·59-s + 8·61-s + 6·67-s − 12·71-s − 2·73-s − 4·77-s − 12·79-s − 10·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 1.20·11-s + 1.45·17-s − 0.917·19-s − 0.834·23-s − 25-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.56·59-s + 1.02·61-s + 0.733·67-s − 1.42·71-s − 0.234·73-s − 0.455·77-s − 1.35·79-s − 1.05·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.10004956995254, −13.83108959374891, −13.01360595206472, −12.83807425949638, −12.10799801524237, −11.84918692588232, −11.07927991293742, −10.76591364594432, −10.06941552158115, −9.865143756888167, −9.254826229967097, −8.306476814594968, −8.246705989553354, −7.554467587400933, −7.333429861974663, −6.318015373929999, −5.784995729329118, −5.633409370774214, −4.649073122627502, −4.368980013256358, −3.646094354237392, −2.853977329208609, −2.432127453579718, −1.681533749892462, −0.8736509745733435, 0,
0.8736509745733435, 1.681533749892462, 2.432127453579718, 2.853977329208609, 3.646094354237392, 4.368980013256358, 4.649073122627502, 5.633409370774214, 5.784995729329118, 6.318015373929999, 7.333429861974663, 7.554467587400933, 8.246705989553354, 8.306476814594968, 9.254826229967097, 9.865143756888167, 10.06941552158115, 10.76591364594432, 11.07927991293742, 11.84918692588232, 12.10799801524237, 12.83807425949638, 13.01360595206472, 13.83108959374891, 14.10004956995254