| L(s) = 1 | + 3·4-s + 4·13-s + 5·16-s − 14·19-s − 3·25-s − 6·31-s − 6·37-s + 16·43-s + 12·52-s + 16·61-s + 3·64-s − 4·67-s − 42·76-s − 8·79-s + 24·97-s − 9·100-s + 26·103-s + 18·109-s − 15·121-s − 18·124-s + 127-s + 131-s + 137-s + 139-s − 18·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 1.10·13-s + 5/4·16-s − 3.21·19-s − 3/5·25-s − 1.07·31-s − 0.986·37-s + 2.43·43-s + 1.66·52-s + 2.04·61-s + 3/8·64-s − 0.488·67-s − 4.81·76-s − 0.900·79-s + 2.43·97-s − 0.899·100-s + 2.56·103-s + 1.72·109-s − 1.36·121-s − 1.61·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.47·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.907642736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.907642736\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−10.10298622004518825749087570464, −9.302885658540947908315063634495, −8.848276503002121803923774537655, −8.725418989418723089930402407623, −8.214090289079762875375570043870, −7.74443858311826301435970244497, −7.35744070570776281569178494331, −6.84416069498005048775938673381, −6.61972073495258015631834214141, −6.07355209842237419310298739082, −5.89795357588904636549185706215, −5.48769918369145578724784166699, −4.55752085290803469902278297907, −4.26299173805444489815711329126, −3.70742956508013399152369602234, −3.27482901584446912986752730007, −2.49286076782179912292461798588, −1.93156615238358713005091899982, −1.90129567862797964692328185767, −0.66453989430569328623377113240,
0.66453989430569328623377113240, 1.90129567862797964692328185767, 1.93156615238358713005091899982, 2.49286076782179912292461798588, 3.27482901584446912986752730007, 3.70742956508013399152369602234, 4.26299173805444489815711329126, 4.55752085290803469902278297907, 5.48769918369145578724784166699, 5.89795357588904636549185706215, 6.07355209842237419310298739082, 6.61972073495258015631834214141, 6.84416069498005048775938673381, 7.35744070570776281569178494331, 7.74443858311826301435970244497, 8.214090289079762875375570043870, 8.725418989418723089930402407623, 8.848276503002121803923774537655, 9.302885658540947908315063634495, 10.10298622004518825749087570464
