| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s + 4·10-s − 10·13-s − 4·16-s + 10·17-s + 4·20-s − 25-s − 20·26-s − 8·32-s + 20·34-s + 14·37-s − 20·41-s − 2·50-s − 20·52-s + 18·53-s + 20·61-s − 8·64-s − 20·65-s + 20·68-s + 10·73-s + 28·74-s − 8·80-s − 9·81-s − 40·82-s + 20·85-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s + 1.26·10-s − 2.77·13-s − 16-s + 2.42·17-s + 0.894·20-s − 1/5·25-s − 3.92·26-s − 1.41·32-s + 3.42·34-s + 2.30·37-s − 3.12·41-s − 0.282·50-s − 2.77·52-s + 2.47·53-s + 2.56·61-s − 64-s − 2.48·65-s + 2.42·68-s + 1.17·73-s + 3.25·74-s − 0.894·80-s − 81-s − 4.41·82-s + 2.16·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.468557847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.468557847\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−10.07777529203296053239248113589, −9.995895005827938782847657328128, −9.630449055576491865947824382297, −9.086489278559170336108820841607, −8.523325819574670859769689729456, −7.961926639505284974667826140339, −7.54973587624073372361618498502, −7.16140182094123318922567456520, −6.73667846196512459043785934601, −6.26070345883133105806874898042, −5.54395070645542367514017902965, −5.33941418353216564672989958363, −5.26653230877033484086553748573, −4.51687053298683043767101964001, −4.11372971456978406634294044017, −3.28457437384958610690493771457, −3.07478254024297055431198054974, −2.19804625517662567576110363493, −2.11859904984605800539185865731, −0.75832861780128527288788226820,
0.75832861780128527288788226820, 2.11859904984605800539185865731, 2.19804625517662567576110363493, 3.07478254024297055431198054974, 3.28457437384958610690493771457, 4.11372971456978406634294044017, 4.51687053298683043767101964001, 5.26653230877033484086553748573, 5.33941418353216564672989958363, 5.54395070645542367514017902965, 6.26070345883133105806874898042, 6.73667846196512459043785934601, 7.16140182094123318922567456520, 7.54973587624073372361618498502, 7.961926639505284974667826140339, 8.523325819574670859769689729456, 9.086489278559170336108820841607, 9.630449055576491865947824382297, 9.995895005827938782847657328128, 10.07777529203296053239248113589
