| L(s) = 1 | − 40·25-s + 56·49-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-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 + 241-s + ⋯ |
| L(s) = 1 | − 8·25-s + 8·49-s − 4·121-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 + 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.945096751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.945096751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( ( 1 + p T^{2} )^{4} \) |
| 19 | \( ( 1 - p T^{2} )^{4} \) |
| good | 5 | \( ( 1 + p T^{2} )^{8} \) |
| 7 | \( ( 1 - p T^{2} )^{8} \) |
| 13 | \( ( 1 - 7 T^{2} - 120 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 23 T^{2} + 240 T^{4} + 23 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 + p T^{2} )^{8} \) |
| 43 | \( ( 1 - p T^{2} )^{8} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 - 103 T^{2} + 7800 T^{4} - 103 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 - p T^{2} )^{8} \) |
| 71 | \( ( 1 - 67 T^{2} - 552 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 - 139 T^{2} + 13080 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 109 T^{2} + 4992 T^{4} - 109 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 31 T^{2} - 6960 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.37307821788676215199960591400, −2.99525619500179013461861623699, −2.95939563736652677395901390828, −2.78408458280605220181841940106, −2.67688571323049050710653110020, −2.61945232739895613479722508011, −2.48852958374385687561246549992, −2.38119172553117514228671488345, −2.33795129951428205460981606912, −2.14641450627228403128946268609, −2.12940650971654575009921013153, −1.98323766173136007772930153534, −1.93017960363261647183386561467, −1.80100424765510532174729607984, −1.64862461001768516062824284918, −1.54080482326575401819105250747, −1.35613638680366418495357266129, −1.23331505670155620426558959304, −1.02000943263608986425193310825, −0.975413134280614393802005143722, −0.793693308409649463217673520114, −0.53747223767850330091223141646, −0.35544010580295572867287398026, −0.32063942211724448659396539156, −0.14694924778403813915291489496,
0.14694924778403813915291489496, 0.32063942211724448659396539156, 0.35544010580295572867287398026, 0.53747223767850330091223141646, 0.793693308409649463217673520114, 0.975413134280614393802005143722, 1.02000943263608986425193310825, 1.23331505670155620426558959304, 1.35613638680366418495357266129, 1.54080482326575401819105250747, 1.64862461001768516062824284918, 1.80100424765510532174729607984, 1.93017960363261647183386561467, 1.98323766173136007772930153534, 2.12940650971654575009921013153, 2.14641450627228403128946268609, 2.33795129951428205460981606912, 2.38119172553117514228671488345, 2.48852958374385687561246549992, 2.61945232739895613479722508011, 2.67688571323049050710653110020, 2.78408458280605220181841940106, 2.95939563736652677395901390828, 2.99525619500179013461861623699, 3.37307821788676215199960591400
Plot not available for L-functions of degree greater than 10.
