| L(s) = 1 | + 124·19-s − 40·25-s + 64·31-s + 178·49-s − 4·61-s + 4·79-s + 416·109-s + 304·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 206·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 6.52·19-s − 8/5·25-s + 2.06·31-s + 3.63·49-s − 0.0655·61-s + 4/79·79-s + 3.81·109-s + 2.51·121-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 − 1.21·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(13.18014170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.18014170\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 8 p T^{2} + p^{4} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 89 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 152 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 103 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 418 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 31 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 568 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 568 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2009 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1112 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 1394 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4258 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3928 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5522 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8537 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9272 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 9929 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 1528 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 2882 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 10169 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.21438062759510482786683276719, −5.94708946174985058045013498214, −5.81473184084430859636675450310, −5.61462112950724982786437976544, −5.40048490831536428858916961246, −5.21606256385350892610978870699, −5.16241608849150930528975852097, −4.83036610195798083841593859173, −4.53155380859506497815264655145, −4.42180453147960649836156957675, −4.08253635947917157952260224222, −3.75819838196827559920334672383, −3.58020344766317732482899897263, −3.38837375895475496290333927180, −3.18242905278834579060466774321, −2.94895449791701500892313100055, −2.72853579326341594798068331471, −2.52404370638662269982798470188, −2.12437926927513239643605328073, −1.78980819500163831422210000937, −1.48630706301953191401225376768, −1.07858715357387718607577657506, −0.822736455912634815373562912243, −0.817008663883462826980026789393, −0.45428692620403345115813374308,
0.45428692620403345115813374308, 0.817008663883462826980026789393, 0.822736455912634815373562912243, 1.07858715357387718607577657506, 1.48630706301953191401225376768, 1.78980819500163831422210000937, 2.12437926927513239643605328073, 2.52404370638662269982798470188, 2.72853579326341594798068331471, 2.94895449791701500892313100055, 3.18242905278834579060466774321, 3.38837375895475496290333927180, 3.58020344766317732482899897263, 3.75819838196827559920334672383, 4.08253635947917157952260224222, 4.42180453147960649836156957675, 4.53155380859506497815264655145, 4.83036610195798083841593859173, 5.16241608849150930528975852097, 5.21606256385350892610978870699, 5.40048490831536428858916961246, 5.61462112950724982786437976544, 5.81473184084430859636675450310, 5.94708946174985058045013498214, 6.21438062759510482786683276719
