L(s) = 1 | + 4-s − 6·9-s + 4·11-s − 24·29-s + 14·31-s − 6·36-s − 28·41-s + 4·44-s + 13·49-s − 28·59-s + 28·61-s − 64-s − 4·71-s − 22·79-s + 9·81-s + 14·89-s − 24·99-s + 24·109-s − 24·116-s + 26·121-s + 14·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 2·9-s + 1.20·11-s − 4.45·29-s + 2.51·31-s − 36-s − 4.37·41-s + 0.603·44-s + 13/7·49-s − 3.64·59-s + 3.58·61-s − 1/8·64-s − 0.474·71-s − 2.47·79-s + 81-s + 1.48·89-s − 2.41·99-s + 2.29·109-s − 2.22·116-s + 2.36·121-s + 1.25·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.134997821\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.134997821\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 15 T^{2} - 64 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 50 T^{2} - 2829 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.496063414300731290279259296454, −8.204297354555299854805385781259, −7.77570836985041152075369572598, −7.64170337091796259960361270332, −7.27072997061889653830996042287, −7.03464507502967990954147533753, −6.89638452408749512590416418684, −6.54248515544569018388702395408, −6.15513546850204214880593903738, −6.15150914836401806077472135918, −5.67486444821599156887001019560, −5.53576678643747861766465223510, −5.42021458664854185025725445700, −4.90789011547799026148635882731, −4.68804610473596988564689145623, −4.16866772115452463122730823764, −4.02322756502015697176893815719, −3.51770189566314867206876180782, −3.26687968000425660613001941969, −3.13890896507679599036655756324, −2.68797973762886446199521333512, −2.00891221474126650298704887708, −1.96514011204830873215476330558, −1.44984773523033436262610638422, −0.42669489621446936658380505525,
0.42669489621446936658380505525, 1.44984773523033436262610638422, 1.96514011204830873215476330558, 2.00891221474126650298704887708, 2.68797973762886446199521333512, 3.13890896507679599036655756324, 3.26687968000425660613001941969, 3.51770189566314867206876180782, 4.02322756502015697176893815719, 4.16866772115452463122730823764, 4.68804610473596988564689145623, 4.90789011547799026148635882731, 5.42021458664854185025725445700, 5.53576678643747861766465223510, 5.67486444821599156887001019560, 6.15150914836401806077472135918, 6.15513546850204214880593903738, 6.54248515544569018388702395408, 6.89638452408749512590416418684, 7.03464507502967990954147533753, 7.27072997061889653830996042287, 7.64170337091796259960361270332, 7.77570836985041152075369572598, 8.204297354555299854805385781259, 8.496063414300731290279259296454