| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8233429268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8233429268\) |
| \(L(2)\) |
|
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
−8.187037525808313048787365052044, −7.70704469118813221780996523076, −7.54769234103788286074600686655, −7.47909758234650719219270350101, −7.15245021020666141610955162648, −7.12700052844975495279607347007, −6.92558545893966515258125224765, −6.07538726898423923798627880906, −5.98811802724430487159094499332, −5.91511289283228431842614537122, −5.70621133444867641572537011631, −5.02448997658070927284224189854, −5.00284099330827885581388988694, −4.73822578971505967185676606675, −4.48127188121463098281824098756, −3.97767831752866403269335233130, −3.76870962771093580906676995185, −3.25673296809460727083687190440, −3.04383476793492133824274001417, −2.68547007180730724662846216483, −2.35743387254572341603819265958, −2.07121160080971664646636201333, −1.74011961597823973397192436866, −0.833994588421096671216963520256, −0.22099546534468426163830346660,
0.22099546534468426163830346660, 0.833994588421096671216963520256, 1.74011961597823973397192436866, 2.07121160080971664646636201333, 2.35743387254572341603819265958, 2.68547007180730724662846216483, 3.04383476793492133824274001417, 3.25673296809460727083687190440, 3.76870962771093580906676995185, 3.97767831752866403269335233130, 4.48127188121463098281824098756, 4.73822578971505967185676606675, 5.00284099330827885581388988694, 5.02448997658070927284224189854, 5.70621133444867641572537011631, 5.91511289283228431842614537122, 5.98811802724430487159094499332, 6.07538726898423923798627880906, 6.92558545893966515258125224765, 7.12700052844975495279607347007, 7.15245021020666141610955162648, 7.47909758234650719219270350101, 7.54769234103788286074600686655, 7.70704469118813221780996523076, 8.187037525808313048787365052044
