| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s + 4·11-s − 5·13-s − 4·15-s − 3·17-s − 2·19-s − 2·23-s − 7·25-s + 10·27-s + 5·29-s + 4·31-s + 8·33-s + 5·37-s − 10·39-s − 3·41-s − 4·43-s − 6·45-s + 12·47-s + 7·49-s − 6·51-s − 26·53-s − 8·55-s − 4·57-s + 12·59-s − 7·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s + 1.20·11-s − 1.38·13-s − 1.03·15-s − 0.727·17-s − 0.458·19-s − 0.417·23-s − 7/5·25-s + 1.92·27-s + 0.928·29-s + 0.718·31-s + 1.39·33-s + 0.821·37-s − 1.60·39-s − 0.468·41-s − 0.609·43-s − 0.894·45-s + 1.75·47-s + 49-s − 0.840·51-s − 3.57·53-s − 1.07·55-s − 0.529·57-s + 1.56·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253817773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253817773\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.23094340367105253418053398955, −9.825999399775965767087706022946, −9.679972346361291225078580527168, −8.967165319171174049151709566647, −8.858584292226010376783974879840, −8.113596399110900883304571362436, −8.106070699399653521566029415623, −7.49121473442466862572331520487, −7.11334738420316164339917363704, −6.65594426196472605741308538527, −6.23694771964597450098093789645, −5.66476915606323131200315827845, −4.64091410519758782338812617929, −4.40262243147517537495310653861, −4.31734404476913735590355326622, −3.27425859005683860092995635217, −3.20156360415955737004991386498, −2.23065693146653431934752254172, −1.90181985042165875705230291766, −0.68742810757359115413087742785,
0.68742810757359115413087742785, 1.90181985042165875705230291766, 2.23065693146653431934752254172, 3.20156360415955737004991386498, 3.27425859005683860092995635217, 4.31734404476913735590355326622, 4.40262243147517537495310653861, 4.64091410519758782338812617929, 5.66476915606323131200315827845, 6.23694771964597450098093789645, 6.65594426196472605741308538527, 7.11334738420316164339917363704, 7.49121473442466862572331520487, 8.106070699399653521566029415623, 8.113596399110900883304571362436, 8.858584292226010376783974879840, 8.967165319171174049151709566647, 9.679972346361291225078580527168, 9.825999399775965767087706022946, 10.23094340367105253418053398955
