| L(s) = 1 | + 28·19-s + 68·31-s + 194·49-s + 260·61-s + 160·79-s + 196·109-s + 448·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 578·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 1.47·19-s + 2.19·31-s + 3.95·49-s + 4.26·61-s + 2.02·79-s + 1.79·109-s + 3.70·121-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 + 3.42·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.190945361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.190945361\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 224 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 289 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 560 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 176 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 800 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2482 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 770 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 673 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2240 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 1582 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 3920 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 6577 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2914 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 10864 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )^{2}( 1 + 146 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17137 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−7.24845914529468342622250852168, −6.70370347543256590443806310184, −6.66226846612237448348965728284, −6.47072041713408641715902750489, −6.11648930981321834293180103861, −5.79686834783196692910215213436, −5.50477521694918878815404360112, −5.48196539488531696122540471028, −5.45921994544957157861369148042, −4.76820743709061340870524540804, −4.72648809367660328975493558619, −4.48265890245514940491915796223, −4.28226274504488356263888716609, −3.83615956040594936207486671052, −3.52874024704247942383791841188, −3.44649684363635337589631808091, −3.27384795371068798099968291653, −2.59111666076150496945980523348, −2.49274221219668928834850719830, −2.25445787935851589081579259691, −2.08023106676399076684814342289, −1.29324073460812350774489227430, −1.02536836385813590530597792080, −0.76595321914697998349363635259, −0.50216710376623250371565499117,
0.50216710376623250371565499117, 0.76595321914697998349363635259, 1.02536836385813590530597792080, 1.29324073460812350774489227430, 2.08023106676399076684814342289, 2.25445787935851589081579259691, 2.49274221219668928834850719830, 2.59111666076150496945980523348, 3.27384795371068798099968291653, 3.44649684363635337589631808091, 3.52874024704247942383791841188, 3.83615956040594936207486671052, 4.28226274504488356263888716609, 4.48265890245514940491915796223, 4.72648809367660328975493558619, 4.76820743709061340870524540804, 5.45921994544957157861369148042, 5.48196539488531696122540471028, 5.50477521694918878815404360112, 5.79686834783196692910215213436, 6.11648930981321834293180103861, 6.47072041713408641715902750489, 6.66226846612237448348965728284, 6.70370347543256590443806310184, 7.24845914529468342622250852168
