| L(s) = 1 | + 3-s − 7-s + 13-s + 2·19-s − 21-s − 25-s − 27-s + 2·31-s + 2·37-s + 39-s + 2·43-s + 49-s + 2·57-s − 2·61-s − 67-s − 2·73-s − 75-s − 79-s − 81-s − 91-s + 2·93-s − 2·97-s − 4·103-s + 109-s + 2·111-s + 2·121-s + 127-s + ⋯ |
| L(s) = 1 | + 3-s − 7-s + 13-s + 2·19-s − 21-s − 25-s − 27-s + 2·31-s + 2·37-s + 39-s + 2·43-s + 49-s + 2·57-s − 2·61-s − 67-s − 2·73-s − 75-s − 79-s − 81-s − 91-s + 2·93-s − 2·97-s − 4·103-s + 109-s + 2·111-s + 2·121-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.606510341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.606510341\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.677853962000689571077258512864, −9.170672477062316034205995503717, −9.042186843722134281448232793420, −8.559666746269037484949126565118, −8.010529295559822910899513070243, −7.77922638635996294022720573281, −7.49246560253283693970212461781, −7.01151212015942324711392525753, −6.36920046172565192977629468872, −6.12675091943907365783702913901, −5.57084126838916633563934531444, −5.56949854893600664994369288042, −4.42813594897073325934726593499, −4.28065774660506646052382418577, −3.80568205553149369716010253161, −3.05718088586235969492704925960, −2.89681049440655625679880817407, −2.64328669925049778356037693813, −1.59841125263865401988053918924, −0.994463184971430736708565272484,
0.994463184971430736708565272484, 1.59841125263865401988053918924, 2.64328669925049778356037693813, 2.89681049440655625679880817407, 3.05718088586235969492704925960, 3.80568205553149369716010253161, 4.28065774660506646052382418577, 4.42813594897073325934726593499, 5.56949854893600664994369288042, 5.57084126838916633563934531444, 6.12675091943907365783702913901, 6.36920046172565192977629468872, 7.01151212015942324711392525753, 7.49246560253283693970212461781, 7.77922638635996294022720573281, 8.010529295559822910899513070243, 8.559666746269037484949126565118, 9.042186843722134281448232793420, 9.170672477062316034205995503717, 9.677853962000689571077258512864
