| L(s) = 1 | − 4·13-s − 25-s + 8·31-s − 12·37-s + 16·43-s − 10·49-s − 12·61-s − 16·67-s + 12·73-s − 8·79-s + 12·97-s − 16·103-s − 20·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.10·13-s − 1/5·25-s + 1.43·31-s − 1.97·37-s + 2.43·43-s − 1.42·49-s − 1.53·61-s − 1.95·67-s + 1.40·73-s − 0.900·79-s + 1.21·97-s − 1.57·103-s − 1.91·109-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−8.305777236497443070782534038490, −7.71329275402156567470165746108, −7.41690505093324189912719380424, −6.99501538087453616784334179820, −6.38630308579687055413134181452, −6.02865089013153653168839127385, −5.44679955410706631858039571597, −4.88259272422500670901481703819, −4.54817272396579411843012119579, −3.96435605934167431298704963621, −3.22572062031461984566707782857, −2.74014153651728842925549978023, −2.09220771280158455955998898349, −1.25110772659726207072734822662, 0,
1.25110772659726207072734822662, 2.09220771280158455955998898349, 2.74014153651728842925549978023, 3.22572062031461984566707782857, 3.96435605934167431298704963621, 4.54817272396579411843012119579, 4.88259272422500670901481703819, 5.44679955410706631858039571597, 6.02865089013153653168839127385, 6.38630308579687055413134181452, 6.99501538087453616784334179820, 7.41690505093324189912719380424, 7.71329275402156567470165746108, 8.305777236497443070782534038490
