| L(s) = 1 | − 2-s + 6·5-s + 8-s − 6·10-s + 6·11-s − 13-s − 16-s − 3·17-s − 7·19-s − 6·22-s + 18·23-s + 17·25-s + 26-s + 3·29-s + 8·31-s + 3·34-s + 37-s + 7·38-s + 6·40-s − 3·41-s + 43-s − 18·46-s − 17·50-s + 3·53-s + 36·55-s − 3·58-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 2.68·5-s + 0.353·8-s − 1.89·10-s + 1.80·11-s − 0.277·13-s − 1/4·16-s − 0.727·17-s − 1.60·19-s − 1.27·22-s + 3.75·23-s + 17/5·25-s + 0.196·26-s + 0.557·29-s + 1.43·31-s + 0.514·34-s + 0.164·37-s + 1.13·38-s + 0.948·40-s − 0.468·41-s + 0.152·43-s − 2.65·46-s − 2.40·50-s + 0.412·53-s + 4.85·55-s − 0.393·58-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.242590675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.242590675\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−8.996795123489523937292537621138, −8.994942055761353529897793517947, −8.478800719826549423130871374049, −8.249127449229970985895755811609, −7.21931037219834221361470092867, −7.15830515947852751820671674706, −6.61800540471677300778273514118, −6.41038279175996214126638083206, −6.16728028892008344265010391605, −5.73490846078038708772837271758, −5.07578500665887372980434998798, −4.77525590125806992299219712494, −4.57993661655689239492855506313, −3.85077819099246777347765305773, −3.15130343657822466505286283482, −2.73411221058803828276500954478, −2.04936540956762329067166001630, −1.96783547546123729022276195676, −1.07132404484260927136148933198, −0.951670530352665084258329685719,
0.951670530352665084258329685719, 1.07132404484260927136148933198, 1.96783547546123729022276195676, 2.04936540956762329067166001630, 2.73411221058803828276500954478, 3.15130343657822466505286283482, 3.85077819099246777347765305773, 4.57993661655689239492855506313, 4.77525590125806992299219712494, 5.07578500665887372980434998798, 5.73490846078038708772837271758, 6.16728028892008344265010391605, 6.41038279175996214126638083206, 6.61800540471677300778273514118, 7.15830515947852751820671674706, 7.21931037219834221361470092867, 8.249127449229970985895755811609, 8.478800719826549423130871374049, 8.994942055761353529897793517947, 8.996795123489523937292537621138
