| L(s) = 1 | + 6·9-s − 20·25-s + 4·49-s − 40·73-s + 27·81-s − 56·97-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 120·225-s + ⋯ |
| L(s) = 1 | + 2·9-s − 4·25-s + 4/7·49-s − 4.68·73-s + 3·81-s − 5.68·97-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s − 8·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.630802886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630802886\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
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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
−7.25262851361504783896179659909, −7.13540129664529171036333224101, −7.08100181402466285672416995097, −6.95956252353826936079512097918, −6.52019909053280814191114002591, −6.11256316522981164088914181311, −6.00884931725512877115327692953, −5.82195847638298504143501597175, −5.61409552938894761200238742026, −5.41006058619133677801391830768, −5.01717303883842896772106303660, −4.55296030389065355570238880970, −4.43247110074070987928508827188, −4.40323874543549743438045570322, −3.86836922592150014121603379110, −3.80607652190433463268422928109, −3.74309387477081638887245261975, −3.09756281512473601456889499708, −2.75816835518076104426723789851, −2.62523968266781872014428847440, −1.86883732524368011485336623159, −1.78081088725830406099711628756, −1.67924139145711383148454219131, −1.10152930121939473163336821913, −0.33364763152082249397841607729,
0.33364763152082249397841607729, 1.10152930121939473163336821913, 1.67924139145711383148454219131, 1.78081088725830406099711628756, 1.86883732524368011485336623159, 2.62523968266781872014428847440, 2.75816835518076104426723789851, 3.09756281512473601456889499708, 3.74309387477081638887245261975, 3.80607652190433463268422928109, 3.86836922592150014121603379110, 4.40323874543549743438045570322, 4.43247110074070987928508827188, 4.55296030389065355570238880970, 5.01717303883842896772106303660, 5.41006058619133677801391830768, 5.61409552938894761200238742026, 5.82195847638298504143501597175, 6.00884931725512877115327692953, 6.11256316522981164088914181311, 6.52019909053280814191114002591, 6.95956252353826936079512097918, 7.08100181402466285672416995097, 7.13540129664529171036333224101, 7.25262851361504783896179659909
