| L(s) = 1 | + 2-s − 3-s + 2·4-s − 6-s + 4·7-s + 5·8-s − 5·11-s − 2·12-s − 5·13-s + 4·14-s + 5·16-s − 2·17-s − 6·19-s − 4·21-s − 5·22-s − 3·23-s − 5·24-s − 5·26-s + 27-s + 8·28-s + 10·29-s − 20·31-s + 10·32-s + 5·33-s − 2·34-s − 37-s − 6·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 4-s − 0.408·6-s + 1.51·7-s + 1.76·8-s − 1.50·11-s − 0.577·12-s − 1.38·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s − 1.37·19-s − 0.872·21-s − 1.06·22-s − 0.625·23-s − 1.02·24-s − 0.980·26-s + 0.192·27-s + 1.51·28-s + 1.85·29-s − 3.59·31-s + 1.76·32-s + 0.870·33-s − 0.342·34-s − 0.164·37-s − 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.031937724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.031937724\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.34319786895825587252796457864, −10.24419886895313053980284995971, −9.485237248828407938583144435844, −8.702230081896554370459201834740, −8.591470480169802219803107256599, −7.80614603724812077913856877935, −7.65363514054744081119178885384, −7.30641428246103330295807043096, −6.94814803986455881243390362418, −6.32470707847133114070957349096, −5.57445227565107637015102221201, −5.54526575162994171369130331891, −4.80460445454356114115074683906, −4.75140025235797832990178209138, −4.17849691018375613480097117200, −3.63069883962512031861749259261, −2.45380879250618343727315700000, −2.23178629504670604827422902644, −1.99141963587262374541710948812, −0.71596842434145240235712128319,
0.71596842434145240235712128319, 1.99141963587262374541710948812, 2.23178629504670604827422902644, 2.45380879250618343727315700000, 3.63069883962512031861749259261, 4.17849691018375613480097117200, 4.75140025235797832990178209138, 4.80460445454356114115074683906, 5.54526575162994171369130331891, 5.57445227565107637015102221201, 6.32470707847133114070957349096, 6.94814803986455881243390362418, 7.30641428246103330295807043096, 7.65363514054744081119178885384, 7.80614603724812077913856877935, 8.591470480169802219803107256599, 8.702230081896554370459201834740, 9.485237248828407938583144435844, 10.24419886895313053980284995971, 10.34319786895825587252796457864
