| L(s) = 1 | − 2·3-s − 4·4-s − 6·5-s − 3·9-s − 11-s + 8·12-s + 12·15-s + 12·16-s + 24·20-s + 6·23-s + 17·25-s + 14·27-s − 10·31-s + 2·33-s + 12·36-s + 22·37-s + 4·44-s + 18·45-s − 24·48-s − 12·53-s + 6·55-s + 18·59-s − 48·60-s − 32·64-s + 10·67-s − 12·69-s + 18·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2·4-s − 2.68·5-s − 9-s − 0.301·11-s + 2.30·12-s + 3.09·15-s + 3·16-s + 5.36·20-s + 1.25·23-s + 17/5·25-s + 2.69·27-s − 1.79·31-s + 0.348·33-s + 2·36-s + 3.61·37-s + 0.603·44-s + 2.68·45-s − 3.46·48-s − 1.64·53-s + 0.809·55-s + 2.34·59-s − 6.19·60-s − 4·64-s + 1.22·67-s − 1.44·69-s + 2.13·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3195731 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3195731 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | | \( 1 \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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
−7.44927930218322722277526119885, −7.05102888697694657596792316990, −6.35946393928988064816172947330, −5.98016833623442124300848518324, −5.33457363303989968032950767365, −5.22586837194321912962810216772, −4.77627871815223419331599351974, −4.26760634035281585123892442853, −4.06364144920564711800863579864, −3.37588925165810734532381222346, −3.31516413382692155506861780875, −2.51077378003816655346374440626, −0.77148952963174533854789924573, −0.70813066257818517183339634810, 0,
0.70813066257818517183339634810, 0.77148952963174533854789924573, 2.51077378003816655346374440626, 3.31516413382692155506861780875, 3.37588925165810734532381222346, 4.06364144920564711800863579864, 4.26760634035281585123892442853, 4.77627871815223419331599351974, 5.22586837194321912962810216772, 5.33457363303989968032950767365, 5.98016833623442124300848518324, 6.35946393928988064816172947330, 7.05102888697694657596792316990, 7.44927930218322722277526119885
