| L(s) = 1 | − 2-s + 3-s − 6-s + 8·7-s + 8-s + 3·9-s + 6·11-s + 2·13-s − 8·14-s − 16-s − 6·17-s − 3·18-s − 7·19-s + 8·21-s − 6·22-s − 6·23-s + 24-s − 2·26-s + 8·27-s + 4·31-s + 6·33-s + 6·34-s + 20·37-s + 7·38-s + 2·39-s − 9·41-s − 8·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s + 3.02·7-s + 0.353·8-s + 9-s + 1.80·11-s + 0.554·13-s − 2.13·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s − 1.60·19-s + 1.74·21-s − 1.27·22-s − 1.25·23-s + 0.204·24-s − 0.392·26-s + 1.53·27-s + 0.718·31-s + 1.04·33-s + 1.02·34-s + 3.28·37-s + 1.13·38-s + 0.320·39-s − 1.40·41-s − 1.23·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.189588774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.189588774\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−10.09474199711844011687907176169, −9.946850356494535096082866129794, −9.149570185205668476990639362047, −8.874323509406739257606047355231, −8.519249016691948735719943177521, −8.345774465162069181527750648356, −7.81236668075384802346719820142, −7.69992835747480889570578386681, −6.72485337033489874047842306545, −6.68664673243537788932100393543, −6.15593470953801047883776310921, −5.36774904382666829579776527488, −4.65934366186294047675663120380, −4.33593392557876322772710668377, −4.30568778485884231597679763653, −3.73474008897104265396212364368, −2.30369861584046849330064066884, −2.19508762073766577431119049007, −1.39598196030327509443736632745, −1.12913565088584557117681843960,
1.12913565088584557117681843960, 1.39598196030327509443736632745, 2.19508762073766577431119049007, 2.30369861584046849330064066884, 3.73474008897104265396212364368, 4.30568778485884231597679763653, 4.33593392557876322772710668377, 4.65934366186294047675663120380, 5.36774904382666829579776527488, 6.15593470953801047883776310921, 6.68664673243537788932100393543, 6.72485337033489874047842306545, 7.69992835747480889570578386681, 7.81236668075384802346719820142, 8.345774465162069181527750648356, 8.519249016691948735719943177521, 8.874323509406739257606047355231, 9.149570185205668476990639362047, 9.946850356494535096082866129794, 10.09474199711844011687907176169
