| L(s) = 1 | − 2·7-s + 4·9-s − 12·17-s + 12·23-s + 8·25-s + 8·31-s + 12·41-s + 3·49-s − 8·63-s + 4·73-s − 16·79-s + 7·81-s + 12·89-s − 20·97-s + 8·103-s − 24·113-s + 24·119-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·153-s + 157-s − 24·161-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 4/3·9-s − 2.91·17-s + 2.50·23-s + 8/5·25-s + 1.43·31-s + 1.87·41-s + 3/7·49-s − 1.00·63-s + 0.468·73-s − 1.80·79-s + 7/9·81-s + 1.27·89-s − 2.03·97-s + 0.788·103-s − 2.25·113-s + 2.20·119-s + 1.27·121-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 − 3.88·153-s + 0.0798·157-s − 1.89·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.357775030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.357775030\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−12.51588423914814900593157577118, −12.27126300408649077822253432492, −11.20963145106593658923445934879, −11.12255650628312117844483794876, −10.66081107345877093107341662982, −10.11442272039905946568840968797, −9.485667278879334692090732005374, −9.004911751428461470183110882247, −8.851605056236298572716769757105, −8.086871306658627060278864183099, −7.15005841305361122506985563705, −6.92141899467840507498312240140, −6.65498718578400222709215122662, −5.95389790471478450676366667013, −4.86066742606486458578622270517, −4.62424908970589260190327034696, −4.04251521556874733912327080382, −2.96682654562792142869145599782, −2.44152844814487206043791525405, −1.07293226290299731109033098120,
1.07293226290299731109033098120, 2.44152844814487206043791525405, 2.96682654562792142869145599782, 4.04251521556874733912327080382, 4.62424908970589260190327034696, 4.86066742606486458578622270517, 5.95389790471478450676366667013, 6.65498718578400222709215122662, 6.92141899467840507498312240140, 7.15005841305361122506985563705, 8.086871306658627060278864183099, 8.851605056236298572716769757105, 9.004911751428461470183110882247, 9.485667278879334692090732005374, 10.11442272039905946568840968797, 10.66081107345877093107341662982, 11.12255650628312117844483794876, 11.20963145106593658923445934879, 12.27126300408649077822253432492, 12.51588423914814900593157577118
