| L(s) = 1 | + 2·3-s − 23·9-s − 92·11-s + 88·13-s − 176·23-s + 146·25-s − 100·27-s − 184·33-s + 664·37-s + 176·39-s − 768·47-s + 582·49-s + 1.26e3·59-s + 472·61-s − 352·69-s + 1.36e3·71-s − 844·73-s + 292·75-s + 421·81-s − 372·83-s − 2.12e3·97-s + 2.11e3·99-s − 468·107-s − 1.44e3·109-s + 1.32e3·111-s − 2.02e3·117-s + 3.68e3·121-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 0.851·9-s − 2.52·11-s + 1.87·13-s − 1.59·23-s + 1.16·25-s − 0.712·27-s − 0.970·33-s + 2.95·37-s + 0.722·39-s − 2.38·47-s + 1.69·49-s + 2.78·59-s + 0.990·61-s − 0.614·69-s + 2.27·71-s − 1.35·73-s + 0.449·75-s + 0.577·81-s − 0.491·83-s − 2.22·97-s + 2.14·99-s − 0.422·107-s − 1.27·109-s + 1.13·111-s − 1.59·117-s + 2.76·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.889754148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.889754148\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 146 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 582 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 46 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 44 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9410 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 13614 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 88 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 16222 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 13718 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 332 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 101774 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 103998 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 384 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 87154 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 630 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 236 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 598926 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 680 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 422 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 431862 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 186 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 490994 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1062 T + p^{3} T^{2} )^{2} \) |
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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
−10.14627806456247866989811847733, −9.722587412219293882320607072647, −9.371178982719582308038554189016, −8.549443018775986474917922113606, −8.417712026496084772638955876846, −8.004048348031983945845953341239, −7.944106313149737029969810731930, −7.16241095760900855771604453791, −6.60152228991979677105754694923, −6.08578241169597877145402428985, −5.68146899919632728924892940304, −5.34563605980899275150093358057, −4.84087923832633892401189048766, −3.94896840718738884857155437326, −3.81427943031934467337686826535, −2.77777746781765632361797862014, −2.73927884104140245989371720745, −2.10483251049161411085291648212, −1.09301814816847048925147489167, −0.39853686812500331513673711474,
0.39853686812500331513673711474, 1.09301814816847048925147489167, 2.10483251049161411085291648212, 2.73927884104140245989371720745, 2.77777746781765632361797862014, 3.81427943031934467337686826535, 3.94896840718738884857155437326, 4.84087923832633892401189048766, 5.34563605980899275150093358057, 5.68146899919632728924892940304, 6.08578241169597877145402428985, 6.60152228991979677105754694923, 7.16241095760900855771604453791, 7.944106313149737029969810731930, 8.004048348031983945845953341239, 8.417712026496084772638955876846, 8.549443018775986474917922113606, 9.371178982719582308038554189016, 9.722587412219293882320607072647, 10.14627806456247866989811847733
