| L(s) = 1 | − 18·13-s − 64·16-s + 68·17-s + 16·23-s − 25·25-s − 340·29-s + 304·43-s + 670·49-s + 996·53-s + 884·61-s + 1.28e3·79-s − 1.24e3·101-s + 584·103-s − 3.35e3·107-s − 1.00e3·113-s + 1.76e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.87e3·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.384·13-s − 16-s + 0.970·17-s + 0.145·23-s − 1/5·25-s − 2.17·29-s + 1.07·43-s + 1.95·49-s + 2.58·53-s + 1.85·61-s + 1.82·79-s − 1.22·101-s + 0.558·103-s − 3.02·107-s − 0.835·113-s + 1.32·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.852·169-s + 0.000439·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.066949985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066949985\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 + 18 T + p^{3} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{2} T + p^{3} T^{2} )( 1 + p^{2} T + p^{3} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1762 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2902 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 170 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 33982 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 98170 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 108942 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 152 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 135750 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 498 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 410722 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 442 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 496550 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 547722 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 773410 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 640 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 139570 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 858098 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1301170 T^{2} + p^{6} T^{4} \) |
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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
−10.48734634336302150768360845436, −10.07409562303432240204763553092, −9.608844012204658652994870263179, −9.186116655909957281536298440597, −8.895158825391044265361775690988, −8.341595915822929853885166529230, −7.72254477129375481928337422519, −7.45632471338663538513494448203, −6.95025554773790938445462376708, −6.60855343194454782808346855469, −5.67894481777349550679293239048, −5.57213854288030506812370034182, −5.12432219172865043444608795570, −4.14673643131581614579156513700, −4.02462492364285316168494956175, −3.35963803363536721829494739676, −2.33612420042349675525232604167, −2.30425421359629335461425387184, −1.18717501191717287579680991845, −0.45951098976466576549887983221,
0.45951098976466576549887983221, 1.18717501191717287579680991845, 2.30425421359629335461425387184, 2.33612420042349675525232604167, 3.35963803363536721829494739676, 4.02462492364285316168494956175, 4.14673643131581614579156513700, 5.12432219172865043444608795570, 5.57213854288030506812370034182, 5.67894481777349550679293239048, 6.60855343194454782808346855469, 6.95025554773790938445462376708, 7.45632471338663538513494448203, 7.72254477129375481928337422519, 8.341595915822929853885166529230, 8.895158825391044265361775690988, 9.186116655909957281536298440597, 9.608844012204658652994870263179, 10.07409562303432240204763553092, 10.48734634336302150768360845436
