| L(s) = 1 | + 4-s − 3·16-s + 10·25-s − 32·31-s + 28·49-s − 7·64-s − 64·79-s + 10·100-s + 44·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 28·196-s + 197-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3/4·16-s + 2·25-s − 5.74·31-s + 4·49-s − 7/8·64-s − 7.20·79-s + 100-s + 4·121-s − 2.87·124-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 2·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.550419008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.550419008\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−8.599162636151681436738944962546, −8.022435229870816450059839091539, −7.62034062224646169049964955014, −7.28962303437650691170977806956, −7.19926118234714285303010699952, −7.17167154835871913373277622081, −7.09561377138973209739503783573, −6.67632237648738808642492125597, −6.14079877240387661532996606886, −5.93080179270630561436205455584, −5.71532054223491841891359507798, −5.44446666294323822495351610361, −5.36982727164323878418020054903, −4.94119399733646795821694323111, −4.46151511206089626715724513623, −4.19008235000124739392123937359, −4.12945467347615594568320606748, −3.52764951586358401703680591064, −3.42380786308474253701765530665, −2.82379646482041172845711512995, −2.69995579189970169256514538482, −2.19089763468231543384921930973, −1.63473975408626015583008622351, −1.58583667890859363804656841915, −0.49163806044208751816251991575,
0.49163806044208751816251991575, 1.58583667890859363804656841915, 1.63473975408626015583008622351, 2.19089763468231543384921930973, 2.69995579189970169256514538482, 2.82379646482041172845711512995, 3.42380786308474253701765530665, 3.52764951586358401703680591064, 4.12945467347615594568320606748, 4.19008235000124739392123937359, 4.46151511206089626715724513623, 4.94119399733646795821694323111, 5.36982727164323878418020054903, 5.44446666294323822495351610361, 5.71532054223491841891359507798, 5.93080179270630561436205455584, 6.14079877240387661532996606886, 6.67632237648738808642492125597, 7.09561377138973209739503783573, 7.17167154835871913373277622081, 7.19926118234714285303010699952, 7.28962303437650691170977806956, 7.62034062224646169049964955014, 8.022435229870816450059839091539, 8.599162636151681436738944962546
