| L(s) = 1 | + 4·4-s − 16·7-s + 12·16-s − 32·25-s − 64·28-s + 64·37-s + 208·43-s + 94·49-s + 32·64-s − 208·67-s + 416·79-s − 128·100-s − 248·109-s − 192·112-s − 160·121-s + 127-s + 131-s + 137-s + 139-s + 256·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 380·169-s + 832·172-s + ⋯ |
| L(s) = 1 | + 4-s − 2.28·7-s + 3/4·16-s − 1.27·25-s − 2.28·28-s + 1.72·37-s + 4.83·43-s + 1.91·49-s + 1/2·64-s − 3.10·67-s + 5.26·79-s − 1.27·100-s − 2.27·109-s − 1.71·112-s − 1.32·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.72·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.24·169-s + 4.83·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.715423917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.715423917\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 512 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 590 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 608 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 52 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 3362 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 5330 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5906 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 6914 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 52 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 2144 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 8546 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2}( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 10496 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 10370 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−9.578053506161439439882526896306, −9.506872784552101045731784993452, −9.208590479469739543456344317057, −8.954999517266684480301556938703, −8.499190322599170232933594039604, −7.950292834003885802741715748291, −7.82160738204052593435463160007, −7.54122839384197865419772652932, −7.33827580467804692197832638757, −7.00348277058581787091935621079, −6.50257469335578455465174726993, −6.22724533109283352828902654850, −6.19767625153465680355170461673, −5.90050167021205748787454122921, −5.64196061328522157600082960094, −4.99130214645521398519526447503, −4.68105477127415839002847091602, −3.90956218647474449240570523247, −3.87407045650152377376382153061, −3.59467404789706305638789990584, −2.72946618301696376772331510698, −2.66914301156086103093288430329, −2.39419391649487292826104733910, −1.37417304224821295229264961243, −0.53443547214887323211108638913,
0.53443547214887323211108638913, 1.37417304224821295229264961243, 2.39419391649487292826104733910, 2.66914301156086103093288430329, 2.72946618301696376772331510698, 3.59467404789706305638789990584, 3.87407045650152377376382153061, 3.90956218647474449240570523247, 4.68105477127415839002847091602, 4.99130214645521398519526447503, 5.64196061328522157600082960094, 5.90050167021205748787454122921, 6.19767625153465680355170461673, 6.22724533109283352828902654850, 6.50257469335578455465174726993, 7.00348277058581787091935621079, 7.33827580467804692197832638757, 7.54122839384197865419772652932, 7.82160738204052593435463160007, 7.950292834003885802741715748291, 8.499190322599170232933594039604, 8.954999517266684480301556938703, 9.208590479469739543456344317057, 9.506872784552101045731784993452, 9.578053506161439439882526896306
