L(s) = 1 | − 2-s + 3-s − 4-s − 2·5-s − 6-s − 2·7-s + 3·8-s + 9-s + 2·10-s + 8·11-s − 12-s + 2·14-s − 2·15-s − 16-s − 12·17-s − 18-s + 2·20-s − 2·21-s − 8·22-s + 3·24-s − 25-s + 27-s + 2·28-s + 2·30-s − 5·32-s + 8·33-s + 12·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 2.41·11-s − 0.288·12-s + 0.534·14-s − 0.516·15-s − 1/4·16-s − 2.91·17-s − 0.235·18-s + 0.447·20-s − 0.436·21-s − 1.70·22-s + 0.612·24-s − 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.365·30-s − 0.883·32-s + 1.39·33-s + 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.563603797439101222564149427482, −8.016950795382934872102021140609, −7.25047783802838427330028450541, −6.98124315529609665917654625469, −6.59977960482578183483840418362, −6.33133624731705641469095662668, −5.38339444811226591615339943697, −4.73297605464640683024446964240, −4.13559084050773741974089362056, −3.83186304984969195490795988429, −3.75673255721942880075450899160, −2.57577614046595831546471565190, −1.95743598224390190763232031362, −1.06130826634777654296171966841, 0,
1.06130826634777654296171966841, 1.95743598224390190763232031362, 2.57577614046595831546471565190, 3.75673255721942880075450899160, 3.83186304984969195490795988429, 4.13559084050773741974089362056, 4.73297605464640683024446964240, 5.38339444811226591615339943697, 6.33133624731705641469095662668, 6.59977960482578183483840418362, 6.98124315529609665917654625469, 7.25047783802838427330028450541, 8.016950795382934872102021140609, 8.563603797439101222564149427482