| L(s) = 1 | + 12·7-s − 12·13-s + 16·19-s + 4·25-s + 12·31-s + 16·43-s + 70·49-s − 48·61-s − 8·67-s − 16·73-s + 12·79-s − 144·91-s − 16·97-s + 12·103-s − 14·121-s + 127-s + 131-s + 192·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 58·169-s + 173-s + ⋯ |
| L(s) = 1 | + 4.53·7-s − 3.32·13-s + 3.67·19-s + 4/5·25-s + 2.15·31-s + 2.43·43-s + 10·49-s − 6.14·61-s − 0.977·67-s − 1.87·73-s + 1.35·79-s − 15.0·91-s − 1.62·97-s + 1.18·103-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 16.6·133-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 + 4.46·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.733775690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.733775690\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 22 T^{2} - 45 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 80 T^{2} + 4719 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 70 T^{2} + 2691 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 10 T^{2} - 3381 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 34 T^{2} - 5733 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.23824527050215561235595832344, −5.98689926911637349513995523815, −5.92602961252431501246274945717, −5.40480990785091113826779383564, −5.30213601353271011718890952087, −5.16993578484318477550292621343, −5.01713707151908187644859517059, −4.83033335750273946677494663929, −4.82443391973584725722530411344, −4.45849625037565525227953101959, −4.42370740236249392024971805486, −4.12301927155028501363902123423, −4.08158811279117276452169339635, −3.30646799273708789446378191144, −3.18537184141483798934378732518, −2.88244264454865510229651543277, −2.74072429583959759424480500839, −2.58000902722702861925644719163, −2.31017286581641824795045201633, −1.81124946847842778938680981598, −1.59741269968171144844912422356, −1.47541968731808437580808632263, −1.24731919510546772602852277618, −0.864000817797500094853724897389, −0.45361962788105974048162629925,
0.45361962788105974048162629925, 0.864000817797500094853724897389, 1.24731919510546772602852277618, 1.47541968731808437580808632263, 1.59741269968171144844912422356, 1.81124946847842778938680981598, 2.31017286581641824795045201633, 2.58000902722702861925644719163, 2.74072429583959759424480500839, 2.88244264454865510229651543277, 3.18537184141483798934378732518, 3.30646799273708789446378191144, 4.08158811279117276452169339635, 4.12301927155028501363902123423, 4.42370740236249392024971805486, 4.45849625037565525227953101959, 4.82443391973584725722530411344, 4.83033335750273946677494663929, 5.01713707151908187644859517059, 5.16993578484318477550292621343, 5.30213601353271011718890952087, 5.40480990785091113826779383564, 5.92602961252431501246274945717, 5.98689926911637349513995523815, 6.23824527050215561235595832344
