| L(s) = 1 | + 2·3-s + 3·9-s + 8·19-s − 2·25-s + 4·27-s + 4·29-s − 12·37-s + 16·47-s + 12·53-s + 16·57-s + 24·59-s − 4·75-s + 5·81-s − 8·83-s + 8·87-s + 32·103-s + 20·109-s − 24·111-s + 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 32·141-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.83·19-s − 2/5·25-s + 0.769·27-s + 0.742·29-s − 1.97·37-s + 2.33·47-s + 1.64·53-s + 2.11·57-s + 3.12·59-s − 0.461·75-s + 5/9·81-s − 0.878·83-s + 0.857·87-s + 3.15·103-s + 1.91·109-s − 2.27·111-s + 3.38·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.69·141-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.042303525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.042303525\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 170 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.511935370661187278504192128186, −8.418011186444346369260911153760, −7.49938692915668028334352650544, −7.44061146408725277794517650659, −7.20466612681010170960161840559, −7.00015337809726267298673717355, −6.18276001427499235957862745686, −6.06704327533583042784645675183, −5.32001893782356592288567186917, −5.31345074408887751402861771455, −4.79014443721362819635833070226, −4.23699768785466131205613081818, −3.80889437637996031292392665729, −3.57662919000828043588294845900, −3.11343650719143624076934396206, −2.70879019228950778150149154573, −2.11650779133272070147401075534, −1.93521082680014549546127995982, −0.992448209155866352425862183114, −0.75234923484583762022169548639,
0.75234923484583762022169548639, 0.992448209155866352425862183114, 1.93521082680014549546127995982, 2.11650779133272070147401075534, 2.70879019228950778150149154573, 3.11343650719143624076934396206, 3.57662919000828043588294845900, 3.80889437637996031292392665729, 4.23699768785466131205613081818, 4.79014443721362819635833070226, 5.31345074408887751402861771455, 5.32001893782356592288567186917, 6.06704327533583042784645675183, 6.18276001427499235957862745686, 7.00015337809726267298673717355, 7.20466612681010170960161840559, 7.44061146408725277794517650659, 7.49938692915668028334352650544, 8.418011186444346369260911153760, 8.511935370661187278504192128186
