| L(s) = 1 | + 4·13-s − 4·37-s − 4·73-s − 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-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 + ⋯ |
| L(s) = 1 | + 4·13-s − 4·37-s − 4·73-s − 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8694605244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8694605244\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
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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
−7.83549547061645042121765394623, −7.31779022604041152949830528562, −7.24121539184512390265557339016, −6.99653012123343644775849492610, −6.81254689025469872329129787550, −6.41368655815196284656496659776, −6.28110856756256488084407364861, −6.25047255963373333986948750064, −5.82121470846177135671677441346, −5.50816170288864579214599052022, −5.46105895518190051972033628609, −5.17495173136970089075037723889, −5.00197457637456366638609138068, −4.36013492501150072990291401627, −4.11886355285127020876414119677, −3.95107740013189464419388771609, −3.94681818133471158596379612285, −3.48464899534862767580549396055, −3.04288437003052259917622483598, −3.03630434389988737926753874782, −2.79133556326799713206313610323, −1.79875325280363937898677624188, −1.79867177698340179480444143221, −1.48032047084430832667545811613, −1.09226915015899176819216067128,
1.09226915015899176819216067128, 1.48032047084430832667545811613, 1.79867177698340179480444143221, 1.79875325280363937898677624188, 2.79133556326799713206313610323, 3.03630434389988737926753874782, 3.04288437003052259917622483598, 3.48464899534862767580549396055, 3.94681818133471158596379612285, 3.95107740013189464419388771609, 4.11886355285127020876414119677, 4.36013492501150072990291401627, 5.00197457637456366638609138068, 5.17495173136970089075037723889, 5.46105895518190051972033628609, 5.50816170288864579214599052022, 5.82121470846177135671677441346, 6.25047255963373333986948750064, 6.28110856756256488084407364861, 6.41368655815196284656496659776, 6.81254689025469872329129787550, 6.99653012123343644775849492610, 7.24121539184512390265557339016, 7.31779022604041152949830528562, 7.83549547061645042121765394623
