| L(s) = 1 | − 6·5-s − 6·17-s − 8·19-s + 17·25-s + 8·31-s + 11·49-s + 24·59-s + 14·61-s − 16·67-s + 24·71-s − 10·73-s + 16·79-s + 36·85-s + 48·95-s − 12·101-s + 16·103-s − 5·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 1.45·17-s − 1.83·19-s + 17/5·25-s + 1.43·31-s + 11/7·49-s + 3.12·59-s + 1.79·61-s − 1.95·67-s + 2.84·71-s − 1.17·73-s + 1.80·79-s + 3.90·85-s + 4.92·95-s − 1.19·101-s + 1.57·103-s − 0.454·121-s − 1.60·125-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 − 3.85·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9736903023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9736903023\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 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
−8.677189307334523947668671563372, −8.363337226451050603817472399491, −8.277023210437421579829632390968, −8.234769601686634692742464933378, −7.32751736569034931640156212367, −7.27270882298199483092958874418, −6.71714998488573631124350837565, −6.69564614107282682641650223683, −5.95012603191619208721959744636, −5.58857720982145297832256257370, −4.70711830707886589617803915784, −4.68494692367126008833653161485, −4.14345795301477356084412913527, −3.97240210870153619274509494671, −3.56966940227906811305434824874, −3.01995029873567641473879516278, −2.21823605208598876950204611481, −2.16865445928831136851258342401, −0.73301545130877721427260601641, −0.49334480651321311363649195246,
0.49334480651321311363649195246, 0.73301545130877721427260601641, 2.16865445928831136851258342401, 2.21823605208598876950204611481, 3.01995029873567641473879516278, 3.56966940227906811305434824874, 3.97240210870153619274509494671, 4.14345795301477356084412913527, 4.68494692367126008833653161485, 4.70711830707886589617803915784, 5.58857720982145297832256257370, 5.95012603191619208721959744636, 6.69564614107282682641650223683, 6.71714998488573631124350837565, 7.27270882298199483092958874418, 7.32751736569034931640156212367, 8.234769601686634692742464933378, 8.277023210437421579829632390968, 8.363337226451050603817472399491, 8.677189307334523947668671563372
