L(s) = 1 | + 2·3-s + 3·9-s − 4·19-s + 7·25-s + 4·27-s + 18·29-s + 10·31-s + 20·37-s + 24·47-s − 18·53-s − 8·57-s − 18·59-s + 14·75-s + 5·81-s + 6·83-s + 36·87-s + 20·93-s − 8·103-s − 8·109-s + 40·111-s + 12·113-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 48·141-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 0.917·19-s + 7/5·25-s + 0.769·27-s + 3.34·29-s + 1.79·31-s + 3.28·37-s + 3.50·47-s − 2.47·53-s − 1.05·57-s − 2.34·59-s + 1.61·75-s + 5/9·81-s + 0.658·83-s + 3.85·87-s + 2.07·93-s − 0.788·103-s − 0.766·109-s + 3.79·111-s + 1.12·113-s + 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.04·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.327382774\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.327382774\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.001459039225397070308812829286, −8.852419166544719296225306386758, −8.282087060702713190370508050758, −8.212846295492343883968322783125, −7.59730443176789810781247623848, −7.58494722152220269601368204874, −6.65030428684729264818970089276, −6.62233196555210255726968134397, −6.17343778876993231085065776638, −5.86702366733265572321818160965, −4.92270996042427055340456072908, −4.66511463529891051653281781832, −4.31648877412164219233067957796, −4.14301318316900065635135627308, −3.09280100643736370452388762144, −2.92619016772606804013986592059, −2.64127991746733602747381643297, −2.10302338141850890160574972475, −1.01962032116672229306102701652, −0.962526604874529470469330704873,
0.962526604874529470469330704873, 1.01962032116672229306102701652, 2.10302338141850890160574972475, 2.64127991746733602747381643297, 2.92619016772606804013986592059, 3.09280100643736370452388762144, 4.14301318316900065635135627308, 4.31648877412164219233067957796, 4.66511463529891051653281781832, 4.92270996042427055340456072908, 5.86702366733265572321818160965, 6.17343778876993231085065776638, 6.62233196555210255726968134397, 6.65030428684729264818970089276, 7.58494722152220269601368204874, 7.59730443176789810781247623848, 8.212846295492343883968322783125, 8.282087060702713190370508050758, 8.852419166544719296225306386758, 9.001459039225397070308812829286