| L(s) = 1 | + 5·9-s − 54·11-s + 70·19-s + 240·29-s − 364·31-s + 714·41-s − 470·49-s − 1.68e3·59-s − 476·61-s + 1.41e3·71-s + 1.30e3·79-s − 704·81-s − 1.47e3·89-s − 270·99-s + 924·101-s − 460·109-s − 475·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.61e3·169-s + ⋯ |
| L(s) = 1 | + 5/27·9-s − 1.48·11-s + 0.845·19-s + 1.53·29-s − 2.10·31-s + 2.71·41-s − 1.37·49-s − 3.70·59-s − 0.999·61-s + 2.36·71-s + 1.85·79-s − 0.965·81-s − 1.75·89-s − 0.274·99-s + 0.910·101-s − 0.404·109-s − 0.356·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.64·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.584450749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584450749\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 470 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 27 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3610 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9385 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 35 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 18250 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 182 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 79990 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 357 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 137110 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 200590 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 195050 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 840 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 238 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 389005 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 708 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 760345 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 650 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 328165 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 735 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 602110 T^{2} + p^{6} T^{4} \) |
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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
−11.23890474387156454736286874748, −10.53087313040322828086421049384, −10.41416317741963029634469654433, −9.478853961532843889041616458048, −9.427679497786137879655490961851, −8.950091214810801509417321879533, −7.976674798775612194678367036671, −7.921197876605761506314142459991, −7.56346089322731835121402259282, −6.84268322012524559332285609938, −6.36451455880112868741155560055, −5.66581019935354795711993019512, −5.37493663583854497280831777474, −4.68648807293860201948425349353, −4.29758836507083490149384442211, −3.32878092578397869563929963882, −2.95312755914009202373772203526, −2.22633472186606494185775171978, −1.39008959333081297590883217788, −0.43288256417019129105538695961,
0.43288256417019129105538695961, 1.39008959333081297590883217788, 2.22633472186606494185775171978, 2.95312755914009202373772203526, 3.32878092578397869563929963882, 4.29758836507083490149384442211, 4.68648807293860201948425349353, 5.37493663583854497280831777474, 5.66581019935354795711993019512, 6.36451455880112868741155560055, 6.84268322012524559332285609938, 7.56346089322731835121402259282, 7.921197876605761506314142459991, 7.976674798775612194678367036671, 8.950091214810801509417321879533, 9.427679497786137879655490961851, 9.478853961532843889041616458048, 10.41416317741963029634469654433, 10.53087313040322828086421049384, 11.23890474387156454736286874748
