| L(s) = 1 | − 46·49-s + 188·61-s + 572·109-s + 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 674·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 0.938·49-s + 3.08·61-s + 5.24·109-s + 4·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.98·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + 0.00418·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.26024534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.26024534\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 337 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2}( 1 + 37 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2}( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 2903 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 8542 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2}( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 9743 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−5.93264008216315846298032665080, −5.58581876835847300197025333982, −5.37034308470402612240572814921, −5.34279244182902997116391009543, −5.19923361312179000351890706407, −4.80209450204865659604483186696, −4.54782324798827374827793554097, −4.50190454019373494713178724456, −4.42169944543401636796727770089, −3.85818987536826226159211079341, −3.83855003710016890742890404865, −3.67373528840387712021319033707, −3.45937257096749890330774416533, −2.98118056872602056156038347155, −2.93961829626645716909146666465, −2.81698979389014605247850474303, −2.55558096148661373383515368903, −1.96069696144736408815556682747, −1.82372337993988820668382183030, −1.78370164505753768942937435632, −1.74305719408826898509786758987, −0.904321006735203733279905055684, −0.67128455473596337982465690793, −0.60584884637264698805156582401, −0.43962552538262555995752402170,
0.43962552538262555995752402170, 0.60584884637264698805156582401, 0.67128455473596337982465690793, 0.904321006735203733279905055684, 1.74305719408826898509786758987, 1.78370164505753768942937435632, 1.82372337993988820668382183030, 1.96069696144736408815556682747, 2.55558096148661373383515368903, 2.81698979389014605247850474303, 2.93961829626645716909146666465, 2.98118056872602056156038347155, 3.45937257096749890330774416533, 3.67373528840387712021319033707, 3.83855003710016890742890404865, 3.85818987536826226159211079341, 4.42169944543401636796727770089, 4.50190454019373494713178724456, 4.54782324798827374827793554097, 4.80209450204865659604483186696, 5.19923361312179000351890706407, 5.34279244182902997116391009543, 5.37034308470402612240572814921, 5.58581876835847300197025333982, 5.93264008216315846298032665080
