| L(s) = 1 | + 6·9-s + 18·11-s − 6·19-s − 6·31-s + 24·41-s − 2·49-s − 6·59-s − 42·61-s − 2·79-s + 27·81-s − 18·89-s + 108·99-s + 18·101-s − 34·109-s + 167·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s − 36·171-s + 173-s + ⋯ |
| L(s) = 1 | + 2·9-s + 5.42·11-s − 1.37·19-s − 1.07·31-s + 3.74·41-s − 2/7·49-s − 0.781·59-s − 5.37·61-s − 0.225·79-s + 3·81-s − 1.90·89-s + 10.8·99-s + 1.79·101-s − 3.25·109-s + 15.1·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s − 2.75·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.733406175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.733406175\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 19 T^{2} - 168 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 85 T^{2} + 5016 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 79 T^{2} + 3432 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 21 T + 208 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + T^{2} - 5328 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.39567610911785876185455373326, −6.16971244119224867706545042908, −6.11590173141358113435337909537, −5.99672989420056487113492302290, −5.97195491624406278477028424106, −5.56511287090902854362350791168, −5.08734011940768840149435933541, −4.67949374212584410767829872163, −4.63209127456194910533060345530, −4.51424619130628288636464542214, −4.17849430268801223419486782381, −4.13463337912226915190978335252, −3.99285191515913570089755753887, −3.82911554809754185762173592752, −3.37294636723242062408804669758, −3.33589164147039848152608507416, −3.09279946045007698457949452501, −2.44690010865228849638325640973, −2.21206724306692298601813684845, −2.04581514374558732256405605944, −1.45440763854946693558467265952, −1.28295683676576889501872977007, −1.26530093417000197079636771288, −1.26257477679841488074544912869, −0.38256911161143717681196339599,
0.38256911161143717681196339599, 1.26257477679841488074544912869, 1.26530093417000197079636771288, 1.28295683676576889501872977007, 1.45440763854946693558467265952, 2.04581514374558732256405605944, 2.21206724306692298601813684845, 2.44690010865228849638325640973, 3.09279946045007698457949452501, 3.33589164147039848152608507416, 3.37294636723242062408804669758, 3.82911554809754185762173592752, 3.99285191515913570089755753887, 4.13463337912226915190978335252, 4.17849430268801223419486782381, 4.51424619130628288636464542214, 4.63209127456194910533060345530, 4.67949374212584410767829872163, 5.08734011940768840149435933541, 5.56511287090902854362350791168, 5.97195491624406278477028424106, 5.99672989420056487113492302290, 6.11590173141358113435337909537, 6.16971244119224867706545042908, 6.39567610911785876185455373326
