L(s) = 1 | − 2-s + 5-s − 5·7-s + 8-s − 10-s − 11-s + 2·13-s + 5·14-s − 16-s − 2·17-s − 2·19-s + 22-s + 2·23-s + 5·25-s − 2·26-s − 14·29-s + 3·31-s + 2·34-s − 5·35-s + 2·37-s + 2·38-s + 40-s − 16·41-s − 16·43-s − 2·46-s + 18·49-s − 5·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.447·5-s − 1.88·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.554·13-s + 1.33·14-s − 1/4·16-s − 0.485·17-s − 0.458·19-s + 0.213·22-s + 0.417·23-s + 25-s − 0.392·26-s − 2.59·29-s + 0.538·31-s + 0.342·34-s − 0.845·35-s + 0.328·37-s + 0.324·38-s + 0.158·40-s − 2.49·41-s − 2.43·43-s − 0.294·46-s + 18/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.052390146672062519661759001090, −8.987502993596546539176358375917, −8.408700540777443383651853930341, −8.321986319479272653051598501084, −7.34857698346096956753519994605, −7.30370645280870270006323951119, −6.69692466510951270840828180251, −6.57924639343688977774191652433, −5.84903467467946763459460985751, −5.79738763322551774832633286997, −5.09291535262411962338791200441, −4.64064294034245620264441927974, −3.99626645461810811324423971507, −3.56448089020997238933474716787, −2.98389922548793269725175542013, −2.79286515478845993797517130510, −1.75906827003116032256639727772, −1.45304953084384183940918788091, 0, 0,
1.45304953084384183940918788091, 1.75906827003116032256639727772, 2.79286515478845993797517130510, 2.98389922548793269725175542013, 3.56448089020997238933474716787, 3.99626645461810811324423971507, 4.64064294034245620264441927974, 5.09291535262411962338791200441, 5.79738763322551774832633286997, 5.84903467467946763459460985751, 6.57924639343688977774191652433, 6.69692466510951270840828180251, 7.30370645280870270006323951119, 7.34857698346096956753519994605, 8.321986319479272653051598501084, 8.408700540777443383651853930341, 8.987502993596546539176358375917, 9.052390146672062519661759001090