| L(s) = 1 | − 2·3-s + 22·5-s + 14·7-s + 6·9-s + 36·11-s + 42·13-s − 44·15-s + 8·17-s − 118·19-s − 28·21-s − 104·23-s + 170·25-s − 70·27-s − 56·29-s + 20·31-s − 72·33-s + 308·35-s + 504·37-s − 84·39-s − 544·41-s + 412·43-s + 132·45-s − 500·47-s + 147·49-s − 16·51-s + 268·53-s + 792·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 1.96·5-s + 0.755·7-s + 2/9·9-s + 0.986·11-s + 0.896·13-s − 0.757·15-s + 0.114·17-s − 1.42·19-s − 0.290·21-s − 0.942·23-s + 1.35·25-s − 0.498·27-s − 0.358·29-s + 0.115·31-s − 0.379·33-s + 1.48·35-s + 2.23·37-s − 0.344·39-s − 2.07·41-s + 1.46·43-s + 0.437·45-s − 1.55·47-s + 3/7·49-s − 0.0439·51-s + 0.694·53-s + 1.94·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.391103188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.391103188\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 22 T + 314 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 36 T + 934 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 42 T + 3410 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 7790 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 118 T + 7566 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 104 T + 26126 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 56 T + 21974 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 20 T + 48510 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 504 T + 159110 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 544 T + 193358 T^{2} + 544 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 412 T + 190278 T^{2} - 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 500 T + 258974 T^{2} + 500 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 268 T + 20222 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 198 T + 410926 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 346 T + 429114 T^{2} + 346 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1008 T + 591974 T^{2} - 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1224 T + 980014 T^{2} + 1224 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 716 T + 832326 T^{2} - 716 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 584 T + 997470 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1230 T + 1395886 T^{2} + 1230 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 596 T + 39542 T^{2} - 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 856 T + 1729230 T^{2} + 856 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.69208682957622607215456088584, −14.62369097678198395086982156473, −13.70183136702895011565083919075, −13.57568083890030922734116225317, −12.90599081637783112485134113995, −12.28577501820692028058771394731, −11.36093492642619555779748751093, −11.18771815661628809501836736454, −10.20938931315726945930967332526, −9.918805086527631263020978388908, −9.215236220419674948269368921095, −8.599699858461191113376582736515, −7.86705961569865380978701039721, −6.74330209892946739289782221690, −5.98807373257345593786995817181, −5.95090519735177069393186873214, −4.77722985473024679147146954418, −3.90000700922733022620747990971, −2.19091360647940084488316623269, −1.46236580211204225338695196422,
1.46236580211204225338695196422, 2.19091360647940084488316623269, 3.90000700922733022620747990971, 4.77722985473024679147146954418, 5.95090519735177069393186873214, 5.98807373257345593786995817181, 6.74330209892946739289782221690, 7.86705961569865380978701039721, 8.599699858461191113376582736515, 9.215236220419674948269368921095, 9.918805086527631263020978388908, 10.20938931315726945930967332526, 11.18771815661628809501836736454, 11.36093492642619555779748751093, 12.28577501820692028058771394731, 12.90599081637783112485134113995, 13.57568083890030922734116225317, 13.70183136702895011565083919075, 14.62369097678198395086982156473, 14.69208682957622607215456088584
