| L(s) = 1 | − 3-s + 2·5-s + 4·11-s − 4·13-s − 2·15-s − 2·17-s − 4·19-s − 8·23-s + 5·25-s + 27-s + 12·29-s + 8·31-s − 4·33-s − 6·37-s + 4·39-s − 12·41-s − 8·43-s + 2·51-s + 2·53-s + 8·55-s + 4·57-s + 4·59-s + 2·61-s − 8·65-s − 4·67-s + 8·69-s − 16·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.20·11-s − 1.10·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 25-s + 0.192·27-s + 2.22·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.640·39-s − 1.87·41-s − 1.21·43-s + 0.280·51-s + 0.274·53-s + 1.07·55-s + 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.992·65-s − 0.488·67-s + 0.963·69-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.623859576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.623859576\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.039367629988677164087487822549, −8.737776359311122898755639086259, −8.399601019082814108694557322147, −8.378071164294737511056049593333, −7.46943663465475481208404243750, −7.10804001178237939714353760102, −6.79428250117502901023082638050, −6.36198209477353888538267792560, −6.01934304424610221282938171200, −5.99463758733792859008281269023, −5.05507143730031900372600048070, −4.80503178898650400769114477995, −4.57825883065890271442410121166, −4.07988435367740936243049930139, −3.31510761264317781261436183873, −3.01527285069226032114489024083, −2.20831200954118109411811826450, −1.96913824625723005979695226143, −1.30942057868347325014594778705, −0.45710176361555174325276386548,
0.45710176361555174325276386548, 1.30942057868347325014594778705, 1.96913824625723005979695226143, 2.20831200954118109411811826450, 3.01527285069226032114489024083, 3.31510761264317781261436183873, 4.07988435367740936243049930139, 4.57825883065890271442410121166, 4.80503178898650400769114477995, 5.05507143730031900372600048070, 5.99463758733792859008281269023, 6.01934304424610221282938171200, 6.36198209477353888538267792560, 6.79428250117502901023082638050, 7.10804001178237939714353760102, 7.46943663465475481208404243750, 8.378071164294737511056049593333, 8.399601019082814108694557322147, 8.737776359311122898755639086259, 9.039367629988677164087487822549
