| L(s) = 1 | − 4-s − 2·7-s + 16-s + 6·19-s + 2·25-s + 2·28-s + 6·31-s − 10·37-s − 10·43-s − 3·49-s + 12·61-s − 64-s − 2·67-s − 6·73-s − 6·76-s − 14·79-s + 30·97-s − 2·100-s + 24·103-s + 14·109-s − 2·112-s − 17·121-s − 6·124-s + 127-s + 131-s − 12·133-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s + 1/4·16-s + 1.37·19-s + 2/5·25-s + 0.377·28-s + 1.07·31-s − 1.64·37-s − 1.52·43-s − 3/7·49-s + 1.53·61-s − 1/8·64-s − 0.244·67-s − 0.702·73-s − 0.688·76-s − 1.57·79-s + 3.04·97-s − 1/5·100-s + 2.36·103-s + 1.34·109-s − 0.188·112-s − 1.54·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s − 1.04·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359733393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359733393\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 13 T + p T^{2} ) \) |
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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
−8.007555618428826104242728859583, −7.49850214773930224745669921186, −7.14727326785100013983164955149, −6.68353609725862071234751274526, −6.27846377324631673216759255706, −5.80133561424136186244396129052, −5.20543613114974569546036540080, −4.96992360744408689484552152692, −4.43589661383561091247720235421, −3.75319460357054553986837961106, −3.24766601307650882030591436321, −3.06741153601361037751736769887, −2.16473739668213439074315963594, −1.40030972244161054669622621365, −0.54438636723677456574594616808,
0.54438636723677456574594616808, 1.40030972244161054669622621365, 2.16473739668213439074315963594, 3.06741153601361037751736769887, 3.24766601307650882030591436321, 3.75319460357054553986837961106, 4.43589661383561091247720235421, 4.96992360744408689484552152692, 5.20543613114974569546036540080, 5.80133561424136186244396129052, 6.27846377324631673216759255706, 6.68353609725862071234751274526, 7.14727326785100013983164955149, 7.49850214773930224745669921186, 8.007555618428826104242728859583
