| L(s) = 1 | − 2·3-s − 8·9-s + 36·13-s + 12·19-s + 44·25-s + 22·27-s − 136·31-s − 16·37-s − 72·39-s − 160·43-s + 14·49-s − 24·57-s + 156·61-s − 24·67-s − 32·73-s − 88·75-s − 128·79-s + 7·81-s + 272·93-s − 8·97-s + 352·103-s − 256·109-s + 32·111-s − 288·117-s + 428·121-s + 127-s + 320·129-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 8/9·9-s + 2.76·13-s + 0.631·19-s + 1.75·25-s + 0.814·27-s − 4.38·31-s − 0.432·37-s − 1.84·39-s − 3.72·43-s + 2/7·49-s − 0.421·57-s + 2.55·61-s − 0.358·67-s − 0.438·73-s − 1.17·75-s − 1.62·79-s + 7/81·81-s + 2.92·93-s − 0.0824·97-s + 3.41·103-s − 2.34·109-s + 0.288·111-s − 2.46·117-s + 3.53·121-s + 0.00787·127-s + 2.48·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.252791658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.252791658\) |
| \(L(2)\) |
|
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 | $D_{4}$ | \( 1 + 2 T + 4 p T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 44 T^{2} + 1034 T^{4} - 44 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 428 T^{2} + 74630 T^{4} - 428 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 18 T + 412 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 988 T^{2} + 407046 T^{4} - 988 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 556 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1444 T^{2} + 980166 T^{4} - 1444 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 2972 T^{2} + 3611558 T^{4} - 2972 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 68 T + 3050 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 1382 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 1292 T^{2} + 2832038 T^{4} - 1292 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 80 T + 5046 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 6148 T^{2} + 19144326 T^{4} - 6148 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 20 T^{2} - 13350138 T^{4} + 20 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 3676 T^{2} + 15964266 T^{4} - 3676 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 78 T + 8620 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 12 T + 8566 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 10588 T^{2} + 78813510 T^{4} - 10588 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 5990 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 64 T + 4434 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 13948 T^{2} + 141899946 T^{4} - 13948 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 11468 T^{2} + 120945830 T^{4} - 11468 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 4 T + 18374 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.57565318768758293949524196323, −6.37847323555336310134218421309, −6.08465868122670412874220340848, −6.06884535217847809692076602020, −5.88633731631096404032400102597, −5.51946391629441657036099319677, −5.36441666371487245042925911743, −5.09900293090204251417495373448, −4.98177090484056357443829449732, −4.91077069530276707392431282043, −4.46525961098624645862309398517, −3.83560987195402167273536169880, −3.82337895275984135860501124033, −3.75290016943174236755932470102, −3.55440930984679643674346690759, −3.18471736204843765778102241595, −2.87535308059901843845985028919, −2.86362493856909474514689165073, −2.16246405534583507265889001118, −1.81218406383563215828597147894, −1.78494252183075279278280317611, −1.20878404049214075260013077921, −1.18344849944422077704229639222, −0.55103391291074497158178166433, −0.20210862863536763185766739520,
0.20210862863536763185766739520, 0.55103391291074497158178166433, 1.18344849944422077704229639222, 1.20878404049214075260013077921, 1.78494252183075279278280317611, 1.81218406383563215828597147894, 2.16246405534583507265889001118, 2.86362493856909474514689165073, 2.87535308059901843845985028919, 3.18471736204843765778102241595, 3.55440930984679643674346690759, 3.75290016943174236755932470102, 3.82337895275984135860501124033, 3.83560987195402167273536169880, 4.46525961098624645862309398517, 4.91077069530276707392431282043, 4.98177090484056357443829449732, 5.09900293090204251417495373448, 5.36441666371487245042925911743, 5.51946391629441657036099319677, 5.88633731631096404032400102597, 6.06884535217847809692076602020, 6.08465868122670412874220340848, 6.37847323555336310134218421309, 6.57565318768758293949524196323
