| L(s) = 1 | + 6·3-s − 13·4-s − 6·5-s + 27·9-s − 78·12-s − 36·15-s + 105·16-s + 78·20-s − 348·23-s − 223·25-s + 108·27-s − 40·31-s − 351·36-s − 598·37-s − 162·45-s − 768·47-s + 630·48-s − 674·49-s + 510·53-s + 1.14e3·59-s + 468·60-s − 533·64-s + 92·67-s − 2.08e3·69-s − 1.26e3·71-s − 1.33e3·75-s − 630·80-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.62·4-s − 0.536·5-s + 9-s − 1.87·12-s − 0.619·15-s + 1.64·16-s + 0.872·20-s − 3.15·23-s − 1.78·25-s + 0.769·27-s − 0.231·31-s − 1.62·36-s − 2.65·37-s − 0.536·45-s − 2.38·47-s + 1.89·48-s − 1.96·49-s + 1.32·53-s + 2.51·59-s + 1.00·60-s − 1.04·64-s + 0.167·67-s − 3.64·69-s − 2.10·71-s − 2.05·75-s − 0.880·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 13 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 674 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4391 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9679 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 174 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 45895 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 299 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 45967 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 86374 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 384 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 255 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 570 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 313994 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 46 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 630 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 447362 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 923870 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 517034 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 207 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1615 T + p^{3} T^{2} )^{2} \) |
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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
−10.44748888702367356561408190524, −10.02439858559156490605870728516, −9.749355025458067377938235294118, −9.570329424333197717497073148825, −8.618150571378549762407161570600, −8.551950639227691100753168202260, −7.968909428997051172598890130281, −7.964906821895192122428475834518, −7.08703638501334050506598938349, −6.56629815769287871302681929844, −5.57400050469017863353439691175, −5.47330850313509346915923343416, −4.34846099422718010718515022995, −4.27339770155972591887109625189, −3.61198362029662094355497536651, −3.32212557149885488204316576468, −2.08501555029363445471520242159, −1.61505110491051936795733995041, 0, 0,
1.61505110491051936795733995041, 2.08501555029363445471520242159, 3.32212557149885488204316576468, 3.61198362029662094355497536651, 4.27339770155972591887109625189, 4.34846099422718010718515022995, 5.47330850313509346915923343416, 5.57400050469017863353439691175, 6.56629815769287871302681929844, 7.08703638501334050506598938349, 7.964906821895192122428475834518, 7.968909428997051172598890130281, 8.551950639227691100753168202260, 8.618150571378549762407161570600, 9.570329424333197717497073148825, 9.749355025458067377938235294118, 10.02439858559156490605870728516, 10.44748888702367356561408190524
