| L(s) = 1 | − 4·5-s − 16·19-s + 11·25-s − 16·29-s − 8·31-s + 24·41-s − 49-s + 16·59-s − 12·61-s + 16·79-s + 8·89-s + 64·95-s + 24·101-s − 4·109-s − 22·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + 32·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 3.67·19-s + 11/5·25-s − 2.97·29-s − 1.43·31-s + 3.74·41-s − 1/7·49-s + 2.08·59-s − 1.53·61-s + 1.80·79-s + 0.847·89-s + 6.56·95-s + 2.38·101-s − 0.383·109-s − 2·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4779890189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4779890189\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 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
−8.204950846556536979289426268798, −8.202114794783466616145419020550, −7.58572814827864194059476194669, −7.35761897900818770359524397274, −7.27811960694866264961540900248, −6.60847851729709563504326879697, −6.21355526195421843031011328976, −6.06075311253779056868295015896, −5.49313238358292633904719100518, −5.07765990034819504723366533938, −4.55022549749723506801316460637, −4.14702199047817695435539275451, −3.90725478816503839315019275517, −3.86216482990394631660876407120, −3.20760516573395121618763039713, −2.58003541959055927515434126948, −2.03355944636581295610418996941, −1.92941188939022906466980902041, −0.805567807020697005653205270531, −0.24301156939256425401014478676,
0.24301156939256425401014478676, 0.805567807020697005653205270531, 1.92941188939022906466980902041, 2.03355944636581295610418996941, 2.58003541959055927515434126948, 3.20760516573395121618763039713, 3.86216482990394631660876407120, 3.90725478816503839315019275517, 4.14702199047817695435539275451, 4.55022549749723506801316460637, 5.07765990034819504723366533938, 5.49313238358292633904719100518, 6.06075311253779056868295015896, 6.21355526195421843031011328976, 6.60847851729709563504326879697, 7.27811960694866264961540900248, 7.35761897900818770359524397274, 7.58572814827864194059476194669, 8.202114794783466616145419020550, 8.204950846556536979289426268798
