| L(s) = 1 | + 2·2-s − 3-s − 5·5-s − 2·6-s + 20·7-s − 8·8-s + 27·9-s − 10·10-s − 76·11-s − 17·13-s + 40·14-s + 5·15-s − 16·16-s + 14·17-s + 54·18-s − 98·19-s − 20·21-s − 152·22-s − 276·23-s + 8·24-s − 34·26-s − 80·27-s + 28·29-s + 10·30-s − 130·31-s + 76·33-s + 28·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.192·3-s − 0.447·5-s − 0.136·6-s + 1.07·7-s − 0.353·8-s + 9-s − 0.316·10-s − 2.08·11-s − 0.362·13-s + 0.763·14-s + 0.0860·15-s − 1/4·16-s + 0.199·17-s + 0.707·18-s − 1.18·19-s − 0.207·21-s − 1.47·22-s − 2.50·23-s + 0.0680·24-s − 0.256·26-s − 0.570·27-s + 0.179·29-s + 0.0608·30-s − 0.753·31-s + 0.400·33-s + 0.141·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5605741575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5605741575\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 37 | $C_2$ | \( 1 + p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 26 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 37 T + p^{3} T^{2} )( 1 + 17 T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 38 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 17 T - 1908 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T - 4717 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 98 T + 2745 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 p T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 65 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 68912 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 229 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 316 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 507 T + 108172 T^{2} + 507 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 38 T - 203935 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 350 T - 104481 T^{2} - 350 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 468 T - 81739 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 1160 T + 987689 T^{2} + 1160 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 694 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 44 T - 491103 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 284 T - 491131 T^{2} - 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 102 T - 694565 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 366 T + p^{3} T^{2} )^{2} \) |
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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
−11.38266654010114686936873383665, −10.55665365368291662814276249760, −10.52035632380877993497468736047, −9.931352646322629773969868825400, −9.610609148111263222085245963851, −8.676706721771702113712895398323, −8.203071744416690287107103751701, −7.78382040382761941713804999040, −7.76797114974458515756817503197, −6.87192184258336687098756155915, −6.38672342143803588217823349574, −5.61762803142278032840397579124, −5.32058664593253338727138838617, −4.59784603861374518129979777612, −4.48001896819980996413074502332, −3.74623590152602293344892020407, −3.05097008832524805191747221852, −2.01427184577112611128478273872, −1.82849988411373416768163873627, −0.21826142754073188418282500585,
0.21826142754073188418282500585, 1.82849988411373416768163873627, 2.01427184577112611128478273872, 3.05097008832524805191747221852, 3.74623590152602293344892020407, 4.48001896819980996413074502332, 4.59784603861374518129979777612, 5.32058664593253338727138838617, 5.61762803142278032840397579124, 6.38672342143803588217823349574, 6.87192184258336687098756155915, 7.76797114974458515756817503197, 7.78382040382761941713804999040, 8.203071744416690287107103751701, 8.676706721771702113712895398323, 9.610609148111263222085245963851, 9.931352646322629773969868825400, 10.52035632380877993497468736047, 10.55665365368291662814276249760, 11.38266654010114686936873383665
