| L(s) = 1 | + 4·2-s + 10·4-s + 20·8-s + 35·16-s + 24·23-s − 10·25-s + 56·32-s + 96·46-s − 4·49-s − 40·50-s + 24·53-s + 84·64-s − 16·79-s + 240·92-s − 16·98-s − 100·100-s + 96·106-s − 48·107-s − 40·109-s − 24·113-s + 40·121-s + 127-s + 120·128-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 7.07·8-s + 35/4·16-s + 5.00·23-s − 2·25-s + 9.89·32-s + 14.1·46-s − 4/7·49-s − 5.65·50-s + 3.29·53-s + 21/2·64-s − 1.80·79-s + 25.0·92-s − 1.61·98-s − 10·100-s + 9.32·106-s − 4.64·107-s − 3.83·109-s − 2.25·113-s + 3.63·121-s + 0.0887·127-s + 10.6·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(23.07869801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.07869801\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 88 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.47434382277426184010969386820, −7.03909844704061318053381462248, −6.90163214638191383753857212386, −6.88910330650222965235845248510, −6.87043759280679137682174790671, −6.28442495715401823956566949450, −6.21694042904162484682703684388, −5.68705620820396830090844243138, −5.57272756508628950437021374759, −5.43656664268181940483207931602, −5.08924272246594649996249701653, −5.05981138889256242548694141051, −4.84037393461193360756176021155, −4.19343185181301668597937393421, −4.19095656564932367032523326488, −3.88923450642458968560413928203, −3.86599417797632285484527579714, −3.17402805827084384692097579422, −3.04367145682432147600519099007, −2.76048153591015490199788523495, −2.67686656673647784230340749782, −2.22805605885252604452251272302, −1.59507422869635237095676200081, −1.40782062468486522491108836982, −0.844671356794809025110105068571,
0.844671356794809025110105068571, 1.40782062468486522491108836982, 1.59507422869635237095676200081, 2.22805605885252604452251272302, 2.67686656673647784230340749782, 2.76048153591015490199788523495, 3.04367145682432147600519099007, 3.17402805827084384692097579422, 3.86599417797632285484527579714, 3.88923450642458968560413928203, 4.19095656564932367032523326488, 4.19343185181301668597937393421, 4.84037393461193360756176021155, 5.05981138889256242548694141051, 5.08924272246594649996249701653, 5.43656664268181940483207931602, 5.57272756508628950437021374759, 5.68705620820396830090844243138, 6.21694042904162484682703684388, 6.28442495715401823956566949450, 6.87043759280679137682174790671, 6.88910330650222965235845248510, 6.90163214638191383753857212386, 7.03909844704061318053381462248, 7.47434382277426184010969386820
