| L(s) = 1 | − 4·3-s + 2·4-s − 6·9-s − 8·12-s − 29·16-s + 84·25-s + 76·27-s + 120·31-s − 12·36-s − 40·37-s + 116·48-s − 84·49-s − 92·64-s − 168·67-s − 336·75-s − 109·81-s − 480·93-s + 296·97-s + 168·100-s + 136·103-s + 152·108-s + 160·111-s + 240·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 1/2·4-s − 2/3·9-s − 2/3·12-s − 1.81·16-s + 3.35·25-s + 2.81·27-s + 3.87·31-s − 1/3·36-s − 1.08·37-s + 2.41·48-s − 1.71·49-s − 1.43·64-s − 2.50·67-s − 4.47·75-s − 1.34·81-s − 5.16·93-s + 3.05·97-s + 1.67·100-s + 1.32·103-s + 1.40·108-s + 1.44·111-s + 1.93·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.843871920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.843871920\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2}( 1 + 3 T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 p T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 166 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 498 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{2}( 1 + 54 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 1570 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 3474 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3066 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3818 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5810 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 2022 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 42 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 5850 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 5058 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 11978 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 13330 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 11970 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.115720536581350838572181502782, −7.72307303757410846253454787104, −7.61853625544454035444026268708, −6.97622487362366834517685153799, −6.94372457870807769274338944411, −6.83737207397697500399098337812, −6.47338713524486528421817897006, −6.15097195189452253059240003112, −6.12873561599567231470033137604, −6.05723558002280440820479234604, −5.30640972456877355992491023694, −5.11849252463177757637651871015, −4.99871706358107821208272676169, −4.54384914216436415397823680611, −4.51870337630517113599828259052, −4.43993234178756986609899650125, −3.40926173386547506982743245182, −3.37266882361904690293500604173, −2.83730532132860287924391173826, −2.70084460433748197247188365141, −2.54165852292630514912220552239, −1.79303090525624972553615161798, −1.33801316401957136952610428961, −0.63650292951615646684212584155, −0.50971400987495749886318810710,
0.50971400987495749886318810710, 0.63650292951615646684212584155, 1.33801316401957136952610428961, 1.79303090525624972553615161798, 2.54165852292630514912220552239, 2.70084460433748197247188365141, 2.83730532132860287924391173826, 3.37266882361904690293500604173, 3.40926173386547506982743245182, 4.43993234178756986609899650125, 4.51870337630517113599828259052, 4.54384914216436415397823680611, 4.99871706358107821208272676169, 5.11849252463177757637651871015, 5.30640972456877355992491023694, 6.05723558002280440820479234604, 6.12873561599567231470033137604, 6.15097195189452253059240003112, 6.47338713524486528421817897006, 6.83737207397697500399098337812, 6.94372457870807769274338944411, 6.97622487362366834517685153799, 7.61853625544454035444026268708, 7.72307303757410846253454787104, 8.115720536581350838572181502782
