| L(s) = 1 | − 6·7-s + 2·9-s + 2·11-s + 4·13-s − 2·17-s − 6·19-s − 2·23-s + 14·29-s + 12·37-s + 8·43-s + 14·47-s + 18·49-s + 6·59-s − 2·61-s − 12·63-s + 8·67-s + 6·73-s − 12·77-s + 16·79-s − 5·81-s + 12·89-s − 24·91-s + 22·97-s + 4·99-s − 10·101-s − 10·103-s − 10·109-s + ⋯ |
| L(s) = 1 | − 2.26·7-s + 2/3·9-s + 0.603·11-s + 1.10·13-s − 0.485·17-s − 1.37·19-s − 0.417·23-s + 2.59·29-s + 1.97·37-s + 1.21·43-s + 2.04·47-s + 18/7·49-s + 0.781·59-s − 0.256·61-s − 1.51·63-s + 0.977·67-s + 0.702·73-s − 1.36·77-s + 1.80·79-s − 5/9·81-s + 1.27·89-s − 2.51·91-s + 2.23·97-s + 0.402·99-s − 0.995·101-s − 0.985·103-s − 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.945521021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.945521021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.559984904623230768120056510743, −9.202777442018719474560797104627, −8.882847590088192414502997957351, −8.626973638120059890081391121437, −7.86882957691107954305422945914, −7.79877819103494708854545042448, −6.81502325209710488933902103630, −6.80769583172628534747188827900, −6.29016806193117293938364217018, −6.26327312788276781106289806141, −5.78312796995093100510417287078, −5.04329672371956769339323219259, −4.35000627073342753358568227050, −4.05001816239171832702076295634, −3.82840733122396495336882871672, −3.16192918225020170956696752608, −2.53961247208360371430219396653, −2.31408596923977728867184232745, −1.10895467926714307885772631133, −0.64075104863877162412199913489,
0.64075104863877162412199913489, 1.10895467926714307885772631133, 2.31408596923977728867184232745, 2.53961247208360371430219396653, 3.16192918225020170956696752608, 3.82840733122396495336882871672, 4.05001816239171832702076295634, 4.35000627073342753358568227050, 5.04329672371956769339323219259, 5.78312796995093100510417287078, 6.26327312788276781106289806141, 6.29016806193117293938364217018, 6.80769583172628534747188827900, 6.81502325209710488933902103630, 7.79877819103494708854545042448, 7.86882957691107954305422945914, 8.626973638120059890081391121437, 8.882847590088192414502997957351, 9.202777442018719474560797104627, 9.559984904623230768120056510743
