| L(s) = 1 | + 3·2-s − 3·3-s + 8·4-s − 18·5-s − 9·6-s + 45·8-s − 54·10-s + 36·11-s − 24·12-s + 68·13-s + 54·15-s + 135·16-s + 42·17-s − 124·19-s − 144·20-s + 108·22-s − 135·24-s + 125·25-s + 204·26-s + 27·27-s + 204·29-s + 162·30-s − 160·31-s + 360·32-s − 108·33-s + 126·34-s − 398·37-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.577·3-s + 4-s − 1.60·5-s − 0.612·6-s + 1.98·8-s − 1.70·10-s + 0.986·11-s − 0.577·12-s + 1.45·13-s + 0.929·15-s + 2.10·16-s + 0.599·17-s − 1.49·19-s − 1.60·20-s + 1.04·22-s − 1.14·24-s + 25-s + 1.53·26-s + 0.192·27-s + 1.30·29-s + 0.985·30-s − 0.926·31-s + 1.98·32-s − 0.569·33-s + 0.635·34-s − 1.76·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.276332127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.276332127\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 18 T + 199 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 36 T - 35 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 42 T - 3149 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 124 T + 8517 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 102 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 160 T - 4191 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 398 T + 107751 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 318 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 268 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 240 T - 46223 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 498 T + 99127 T^{2} - 498 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 132 T - 187955 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 398 T - 68577 T^{2} - 398 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 92 T - 292299 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 720 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 502 T - 137013 T^{2} + 502 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1024 T + 555537 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 204 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 354 T - 579653 T^{2} - 354 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 286 T + p^{3} T^{2} )^{2} \) |
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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
−12.71218598603003160721609342051, −12.28697981330904298992745168692, −11.79728383950446893812596426536, −11.57055594512183032906997135319, −10.81465395515391972633576825846, −10.71907371309498578405815356369, −10.20660626282546537536112853442, −9.072115043447400043218390156351, −8.389666447589990346470292606482, −8.152239575144576424737485159807, −7.23993331226180311749015294910, −7.02190256718262818442342952800, −6.23722269931315501419025048534, −5.75745178459349341440121657357, −4.83661135509476751219375943893, −4.23243873122062232317082893201, −3.89271019212947467248293205036, −3.28215871179917428937253436812, −1.82498344050438019689311163934, −0.831499309280959607462589815620,
0.831499309280959607462589815620, 1.82498344050438019689311163934, 3.28215871179917428937253436812, 3.89271019212947467248293205036, 4.23243873122062232317082893201, 4.83661135509476751219375943893, 5.75745178459349341440121657357, 6.23722269931315501419025048534, 7.02190256718262818442342952800, 7.23993331226180311749015294910, 8.152239575144576424737485159807, 8.389666447589990346470292606482, 9.072115043447400043218390156351, 10.20660626282546537536112853442, 10.71907371309498578405815356369, 10.81465395515391972633576825846, 11.57055594512183032906997135319, 11.79728383950446893812596426536, 12.28697981330904298992745168692, 12.71218598603003160721609342051
