| L(s) = 1 | − 4-s − 9-s + 6·11-s + 16-s + 6·19-s − 8·29-s + 2·31-s + 36-s + 8·41-s − 6·44-s + 5·49-s − 12·59-s − 4·61-s − 64-s + 14·71-s − 6·76-s + 2·79-s + 81-s − 2·89-s − 6·99-s − 2·101-s − 40·109-s + 8·116-s + 5·121-s − 2·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1.80·11-s + 1/4·16-s + 1.37·19-s − 1.48·29-s + 0.359·31-s + 1/6·36-s + 1.24·41-s − 0.904·44-s + 5/7·49-s − 1.56·59-s − 0.512·61-s − 1/8·64-s + 1.66·71-s − 0.688·76-s + 0.225·79-s + 1/9·81-s − 0.211·89-s − 0.603·99-s − 0.199·101-s − 3.83·109-s + 0.742·116-s + 5/11·121-s − 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.492453828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.492453828\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.352908837777161780622767911779, −8.164878581881082805386353012400, −7.86762978860795334826875172203, −7.36327569423185384288533927900, −7.06256946951456400994554699672, −6.69828176887307990528225837183, −6.34753415638446266439330950306, −5.77891594023404932690400278135, −5.67434622530103733051830274612, −5.26792528833001663964848914339, −4.71459192377336990997630120274, −4.30247899995065091114816768902, −4.02675842500028177692737931271, −3.55666236049656402587146008709, −3.26041340519396988623860377630, −2.73378393516684869698336643202, −2.14087115378015507587338527226, −1.50807896490172012490261394488, −1.14428449258354657984277874959, −0.49163792157797104475249031551,
0.49163792157797104475249031551, 1.14428449258354657984277874959, 1.50807896490172012490261394488, 2.14087115378015507587338527226, 2.73378393516684869698336643202, 3.26041340519396988623860377630, 3.55666236049656402587146008709, 4.02675842500028177692737931271, 4.30247899995065091114816768902, 4.71459192377336990997630120274, 5.26792528833001663964848914339, 5.67434622530103733051830274612, 5.77891594023404932690400278135, 6.34753415638446266439330950306, 6.69828176887307990528225837183, 7.06256946951456400994554699672, 7.36327569423185384288533927900, 7.86762978860795334826875172203, 8.164878581881082805386353012400, 8.352908837777161780622767911779
