| L(s) = 1 | + 3·2-s + 4·4-s − 9·5-s + 13·7-s + 9·8-s − 27·10-s + 15·11-s + 39·14-s + 27·16-s − 18·17-s − 18·19-s − 36·20-s + 45·22-s + 29·25-s + 52·28-s + 18·29-s − 21·31-s + 36·32-s − 54·34-s − 117·35-s − 10·37-s − 54·38-s − 81·40-s − 148·43-s + 60·44-s + 120·49-s + 87·50-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 4-s − 9/5·5-s + 13/7·7-s + 9/8·8-s − 2.69·10-s + 1.36·11-s + 2.78·14-s + 1.68·16-s − 1.05·17-s − 0.947·19-s − 9/5·20-s + 2.04·22-s + 1.15·25-s + 13/7·28-s + 0.620·29-s − 0.677·31-s + 9/8·32-s − 1.58·34-s − 3.34·35-s − 0.270·37-s − 1.42·38-s − 2.02·40-s − 3.44·43-s + 1.36·44-s + 2.44·49-s + 1.73·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.627033837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.627033837\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 13 T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T + 104 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 18 T + 397 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T + 469 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 21 T + 1108 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 10 T - 1269 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3254 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 33 T - 1720 T^{2} - 33 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 27 T + 3724 T^{2} + 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 156 T + 11833 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 76 T + 1287 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 84 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 108 T + 9217 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 43 T - 4392 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 505 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 126 T + 13213 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 15529 T^{2} + p^{4} T^{4} \) |
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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
−15.02440177443651346987835039344, −14.34179588260537976741302652368, −14.19787186529056856333953852778, −13.22023780044727537780138148508, −13.01467959954359489871181972538, −11.92271286537109320194194371255, −11.76314063414850203315741697117, −11.49097619127482036800611643979, −10.81155921747065354100706126177, −10.20639122071217176756197278874, −8.691417973769764465586934621146, −8.537932563896892917936526300135, −7.76818233545186321744949805850, −7.16378386346551290670226833160, −6.45839266468014701909286460584, −5.18310419694613783631327157725, −4.66848313916328616645734800157, −4.10362818110744039781381680296, −3.67630557137446981860248614035, −1.76505728773770280277152985382,
1.76505728773770280277152985382, 3.67630557137446981860248614035, 4.10362818110744039781381680296, 4.66848313916328616645734800157, 5.18310419694613783631327157725, 6.45839266468014701909286460584, 7.16378386346551290670226833160, 7.76818233545186321744949805850, 8.537932563896892917936526300135, 8.691417973769764465586934621146, 10.20639122071217176756197278874, 10.81155921747065354100706126177, 11.49097619127482036800611643979, 11.76314063414850203315741697117, 11.92271286537109320194194371255, 13.01467959954359489871181972538, 13.22023780044727537780138148508, 14.19787186529056856333953852778, 14.34179588260537976741302652368, 15.02440177443651346987835039344
