| L(s) = 1 | + 4·25-s + 24·29-s − 16·49-s − 8·61-s + 16·79-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 4/5·25-s + 4.45·29-s − 2.28·49-s − 1.02·61-s + 1.80·79-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 + 4/13·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.967831258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.967831258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 + 8 T^{2} + 30 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 p T^{2} + 342 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 28 T^{2} + 438 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 46 T^{2} + 1062 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 58 T^{2} + 1710 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 22 T^{2} + 342 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 80 T^{2} + 3582 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 76 T^{2} + 4470 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 106 T^{2} + 6318 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 56 T^{2} + 4446 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 172 T^{2} + 12678 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 46 T^{2} + 7302 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 200 T^{2} + 18222 T^{4} + 200 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 238 T^{2} + 23718 T^{4} - 238 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 224 T^{2} + 22446 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 56 T^{2} - 4338 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 236 T^{2} + 27366 T^{4} + 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.78348497722495011716508190698, −3.61380271167204722798014458914, −3.52310842110320995232325098249, −3.32054657989191008114347971410, −3.23494085194311837011482541627, −3.11369109157901313184761250681, −2.91550513440507312218223607307, −2.91029734572111028127310882426, −2.85077883068820921034692762396, −2.79457865929768367494497283365, −2.40660720027854565326581310120, −2.38352322610016116914268579209, −2.38301793577606346821246902557, −2.19616252027988919280101403699, −1.84993176096643493807200650899, −1.82269123301707628866674919465, −1.59111012797310201527986228840, −1.52317608448845723499918858089, −1.36152851154637145478810839500, −1.27788278724303885137926953958, −0.868815533471502768895775585339, −0.813956544742581551581483169850, −0.59226558650566471696224689311, −0.58408631946427733050547015846, −0.18130348554301832961021756725,
0.18130348554301832961021756725, 0.58408631946427733050547015846, 0.59226558650566471696224689311, 0.813956544742581551581483169850, 0.868815533471502768895775585339, 1.27788278724303885137926953958, 1.36152851154637145478810839500, 1.52317608448845723499918858089, 1.59111012797310201527986228840, 1.82269123301707628866674919465, 1.84993176096643493807200650899, 2.19616252027988919280101403699, 2.38301793577606346821246902557, 2.38352322610016116914268579209, 2.40660720027854565326581310120, 2.79457865929768367494497283365, 2.85077883068820921034692762396, 2.91029734572111028127310882426, 2.91550513440507312218223607307, 3.11369109157901313184761250681, 3.23494085194311837011482541627, 3.32054657989191008114347971410, 3.52310842110320995232325098249, 3.61380271167204722798014458914, 3.78348497722495011716508190698
Plot not available for L-functions of degree greater than 10.
