| L(s) = 1 | − 3-s − 4·5-s + 9-s − 4·13-s + 4·15-s + 2·17-s + 2·25-s − 27-s + 12·29-s + 16·31-s + 4·39-s − 12·41-s − 4·45-s − 14·49-s − 2·51-s + 8·59-s + 16·65-s − 2·75-s − 16·79-s + 81-s − 8·83-s − 8·85-s − 12·87-s − 16·93-s + 36·113-s − 4·117-s − 6·121-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.10·13-s + 1.03·15-s + 0.485·17-s + 2/5·25-s − 0.192·27-s + 2.22·29-s + 2.87·31-s + 0.640·39-s − 1.87·41-s − 0.596·45-s − 2·49-s − 0.280·51-s + 1.04·59-s + 1.98·65-s − 0.230·75-s − 1.80·79-s + 1/9·81-s − 0.878·83-s − 0.867·85-s − 1.28·87-s − 1.65·93-s + 3.38·113-s − 0.369·117-s − 0.545·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 998784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 998784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6512086180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6512086180\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.098990694093691505068092710868, −7.74878911780238335251144787670, −7.24438582428163300111610536895, −6.86045414165966813923304660414, −6.42897107072744719896577758454, −6.04027405491785134295543211817, −5.21641739768646608654789008027, −4.89120977075902183195495535017, −4.36431822091007852489476792578, −4.25303028692796488061253338187, −3.26035088594232051525892793115, −3.12345153078440868283045421761, −2.31170625009033846632510270378, −1.27040457804671527362299532753, −0.42420891881528178575912901086,
0.42420891881528178575912901086, 1.27040457804671527362299532753, 2.31170625009033846632510270378, 3.12345153078440868283045421761, 3.26035088594232051525892793115, 4.25303028692796488061253338187, 4.36431822091007852489476792578, 4.89120977075902183195495535017, 5.21641739768646608654789008027, 6.04027405491785134295543211817, 6.42897107072744719896577758454, 6.86045414165966813923304660414, 7.24438582428163300111610536895, 7.74878911780238335251144787670, 8.098990694093691505068092710868
