| L(s) = 1 | − 52·7-s − 9·9-s + 12·17-s + 296·23-s − 25·25-s + 196·31-s + 644·41-s − 952·47-s + 1.34e3·49-s + 468·63-s + 1.49e3·71-s + 2.32e3·73-s − 620·79-s + 81·81-s − 980·89-s + 2.33e3·97-s + 2.11e3·103-s − 3.51e3·113-s − 624·119-s + 1.06e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 108·153-s + ⋯ |
| L(s) = 1 | − 2.80·7-s − 1/3·9-s + 0.171·17-s + 2.68·23-s − 1/5·25-s + 1.13·31-s + 2.45·41-s − 2.95·47-s + 3.91·49-s + 0.935·63-s + 2.50·71-s + 3.72·73-s − 0.882·79-s + 1/9·81-s − 1.16·89-s + 2.44·97-s + 2.02·103-s − 2.92·113-s − 0.480·119-s + 0.797·121-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 − 0.0570·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.442163505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.442163505\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1062 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4250 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13702 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 148 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 37658 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 98 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 8890 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 322 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 8470 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 476 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 283830 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 408822 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 447562 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 594470 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 748 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1162 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 310 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 127510 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 490 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1166 T + p^{3} T^{2} )^{2} \) |
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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
−10.81221146958276890219999037950, −10.19550015558394138050066903886, −9.731264922067917154291395353671, −9.632961744282965074547330146565, −8.973889891101352742010430222744, −8.941459008143920262295456266787, −7.930506007637125009815533984774, −7.73031010220982061576386932080, −6.71704685867300701918069384189, −6.67248542525218255007456043378, −6.44969365685989810280548378669, −5.72609929205555750462184821079, −5.15596519191607597264847631728, −4.62794530931952565132260440889, −3.63831042779824770018486137225, −3.43114311150400801973272458921, −2.84302535743211056896166605190, −2.42948734984238144700471620260, −1.00756375656053006301426759580, −0.46254395905598964464502678398,
0.46254395905598964464502678398, 1.00756375656053006301426759580, 2.42948734984238144700471620260, 2.84302535743211056896166605190, 3.43114311150400801973272458921, 3.63831042779824770018486137225, 4.62794530931952565132260440889, 5.15596519191607597264847631728, 5.72609929205555750462184821079, 6.44969365685989810280548378669, 6.67248542525218255007456043378, 6.71704685867300701918069384189, 7.73031010220982061576386932080, 7.930506007637125009815533984774, 8.941459008143920262295456266787, 8.973889891101352742010430222744, 9.632961744282965074547330146565, 9.731264922067917154291395353671, 10.19550015558394138050066903886, 10.81221146958276890219999037950
