L(s) = 1 | − 2-s + 2·3-s − 2·4-s − 2·6-s − 6·7-s + 3·8-s + 3·9-s − 4·12-s + 2·13-s + 6·14-s + 16-s + 4·17-s − 3·18-s − 10·19-s − 12·21-s − 7·23-s + 6·24-s − 2·26-s + 4·27-s + 12·28-s + 5·29-s + 14·31-s − 2·32-s − 4·34-s − 6·36-s + 6·37-s + 10·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s − 4-s − 0.816·6-s − 2.26·7-s + 1.06·8-s + 9-s − 1.15·12-s + 0.554·13-s + 1.60·14-s + 1/4·16-s + 0.970·17-s − 0.707·18-s − 2.29·19-s − 2.61·21-s − 1.45·23-s + 1.22·24-s − 0.392·26-s + 0.769·27-s + 2.26·28-s + 0.928·29-s + 2.51·31-s − 0.353·32-s − 0.685·34-s − 36-s + 0.986·37-s + 1.62·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 57 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 53 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 63 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 111 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 63 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 147 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 21 T + 243 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 207 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 133 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 223 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 223 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.54022256928087440928849863933, −7.51425302664582187731689373083, −6.79718219117286111611503519064, −6.45230954958800040088864973951, −6.33394425404875135764584369103, −6.15759014921972241762739739026, −5.56021677944454389492443518843, −5.09522940414689261137001707323, −4.52953916624470647373469999962, −4.23122651064496257494222640523, −3.93267732805240572372565729720, −3.81311873767280741288616739388, −3.08801349628430757443220575255, −2.93319977592277418477656936437, −2.48609794171826852903278564832, −2.12157413766802336142173917980, −1.23307826625401211238811328081, −0.980101467011135921722989436570, 0, 0,
0.980101467011135921722989436570, 1.23307826625401211238811328081, 2.12157413766802336142173917980, 2.48609794171826852903278564832, 2.93319977592277418477656936437, 3.08801349628430757443220575255, 3.81311873767280741288616739388, 3.93267732805240572372565729720, 4.23122651064496257494222640523, 4.52953916624470647373469999962, 5.09522940414689261137001707323, 5.56021677944454389492443518843, 6.15759014921972241762739739026, 6.33394425404875135764584369103, 6.45230954958800040088864973951, 6.79718219117286111611503519064, 7.51425302664582187731689373083, 7.54022256928087440928849863933