| L(s) = 1 | + 9-s + 4·13-s − 6·17-s − 6·25-s + 12·29-s + 8·37-s − 5·49-s − 12·53-s − 8·61-s − 2·73-s − 8·81-s − 12·89-s − 12·97-s − 12·101-s − 12·109-s + 12·113-s + 4·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 1.10·13-s − 1.45·17-s − 6/5·25-s + 2.22·29-s + 1.31·37-s − 5/7·49-s − 1.64·53-s − 1.02·61-s − 0.234·73-s − 8/9·81-s − 1.27·89-s − 1.21·97-s − 1.19·101-s − 1.14·109-s + 1.12·113-s + 0.369·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.485·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−7.79479479553236507364486493732, −7.27774099213358132015809486803, −6.68957202721806193978273217581, −6.35014987329731172636910476357, −6.18099911021454210276504007249, −5.58353039983297327223378157199, −4.90711614985539699610999865212, −4.52831809305135001637386747581, −4.13546939951366041525236225079, −3.68248419208624114766605593517, −2.86217733411437857444907232853, −2.60478905040558338261948072875, −1.68216410306367556145215742738, −1.19792738995265454495203065562, 0,
1.19792738995265454495203065562, 1.68216410306367556145215742738, 2.60478905040558338261948072875, 2.86217733411437857444907232853, 3.68248419208624114766605593517, 4.13546939951366041525236225079, 4.52831809305135001637386747581, 4.90711614985539699610999865212, 5.58353039983297327223378157199, 6.18099911021454210276504007249, 6.35014987329731172636910476357, 6.68957202721806193978273217581, 7.27774099213358132015809486803, 7.79479479553236507364486493732
