| L(s) = 1 | + 4-s − 3·5-s + 4·7-s − 3·9-s + 3·11-s + 3·13-s − 3·16-s − 3·17-s − 9·19-s − 3·20-s + 9·23-s + 5·25-s + 4·28-s − 9·29-s − 12·35-s − 3·36-s − 7·37-s − 3·41-s − 43-s + 3·44-s + 9·45-s + 9·49-s + 3·52-s + 15·53-s − 9·55-s − 12·63-s − 7·64-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.34·5-s + 1.51·7-s − 9-s + 0.904·11-s + 0.832·13-s − 3/4·16-s − 0.727·17-s − 2.06·19-s − 0.670·20-s + 1.87·23-s + 25-s + 0.755·28-s − 1.67·29-s − 2.02·35-s − 1/2·36-s − 1.15·37-s − 0.468·41-s − 0.152·43-s + 0.452·44-s + 1.34·45-s + 9/7·49-s + 0.416·52-s + 2.06·53-s − 1.21·55-s − 1.51·63-s − 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7803163189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7803163189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.35751208257009218854921448646, −14.70834666301896751611776578217, −14.56036387730836949818660069239, −13.49058812877495325330719049175, −13.21466091278928643433000240031, −12.16359412649119906651873937512, −11.77754183452763774526465769412, −11.28993963185115458104612883538, −10.83313545070113208300751799377, −10.77707182011915899439555582822, −8.959772757758035218124934610671, −8.824342011500182348229732239823, −8.380777143027558262929981240096, −7.49096196586189799249555304195, −6.91967742670604685606408541649, −6.20075781517012772590908988763, −5.09613453097170482446945321184, −4.34580941512952108439354401796, −3.57904951207528452812854647994, −2.07546201997506410131373966669,
2.07546201997506410131373966669, 3.57904951207528452812854647994, 4.34580941512952108439354401796, 5.09613453097170482446945321184, 6.20075781517012772590908988763, 6.91967742670604685606408541649, 7.49096196586189799249555304195, 8.380777143027558262929981240096, 8.824342011500182348229732239823, 8.959772757758035218124934610671, 10.77707182011915899439555582822, 10.83313545070113208300751799377, 11.28993963185115458104612883538, 11.77754183452763774526465769412, 12.16359412649119906651873937512, 13.21466091278928643433000240031, 13.49058812877495325330719049175, 14.56036387730836949818660069239, 14.70834666301896751611776578217, 15.35751208257009218854921448646
