L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s − 4·7-s − 2·9-s − 2·10-s − 2·11-s − 2·12-s − 8·14-s + 15-s − 4·16-s + 4·17-s − 4·18-s − 2·20-s + 4·21-s − 4·22-s − 4·25-s + 5·27-s − 8·28-s + 2·30-s − 8·32-s + 2·33-s + 8·34-s + 4·35-s − 4·36-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 1.51·7-s − 2/3·9-s − 0.632·10-s − 0.603·11-s − 0.577·12-s − 2.13·14-s + 0.258·15-s − 16-s + 0.970·17-s − 0.942·18-s − 0.447·20-s + 0.872·21-s − 0.852·22-s − 4/5·25-s + 0.962·27-s − 1.51·28-s + 0.365·30-s − 1.41·32-s + 0.348·33-s + 1.37·34-s + 0.676·35-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.098969828\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.098969828\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.394238320905654147490500873996, −8.228854358433494441874279371750, −7.49534540311409429819739808064, −6.89747784534263149853829967422, −6.66503987304844816686807720890, −6.10845204313762133714030223921, −5.66407660024727617127120629681, −5.40055166000655653433368694834, −4.88140290332121063087048287423, −4.19085115174357929506261869806, −3.68709007429289538959412107978, −3.09000916592887094247460325417, −2.96905370396904801865789065276, −2.00337434233232539103171654262, −0.44102589544325217691153494662,
0.44102589544325217691153494662, 2.00337434233232539103171654262, 2.96905370396904801865789065276, 3.09000916592887094247460325417, 3.68709007429289538959412107978, 4.19085115174357929506261869806, 4.88140290332121063087048287423, 5.40055166000655653433368694834, 5.66407660024727617127120629681, 6.10845204313762133714030223921, 6.66503987304844816686807720890, 6.89747784534263149853829967422, 7.49534540311409429819739808064, 8.228854358433494441874279371750, 8.394238320905654147490500873996