| L(s) = 1 | − 2-s + 2·5-s + 5·7-s + 8-s − 2·10-s − 3·11-s − 5·13-s − 5·14-s − 16-s − 8·17-s + 5·19-s + 3·22-s − 4·23-s + 3·25-s + 5·26-s − 4·29-s − 4·31-s + 8·34-s + 10·35-s + 7·37-s − 5·38-s + 2·40-s + 6·41-s − 6·43-s + 4·46-s + 6·47-s + 7·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.894·5-s + 1.88·7-s + 0.353·8-s − 0.632·10-s − 0.904·11-s − 1.38·13-s − 1.33·14-s − 1/4·16-s − 1.94·17-s + 1.14·19-s + 0.639·22-s − 0.834·23-s + 3/5·25-s + 0.980·26-s − 0.742·29-s − 0.718·31-s + 1.37·34-s + 1.69·35-s + 1.15·37-s − 0.811·38-s + 0.316·40-s + 0.937·41-s − 0.914·43-s + 0.589·46-s + 0.875·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.389152312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389152312\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
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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
−9.730961028036697187419813221672, −9.694716791708732771998254663060, −9.285166336245128472599885348987, −8.605343572459844553586559349887, −8.453361736168553968633588603822, −8.025891254817114467795355804191, −7.54875238981829342683413107520, −7.15283522626009183354491701957, −7.02843128855991189148146013298, −6.10605778095352686373868321810, −5.65101637436830694972670324613, −5.34512157692928148289493860844, −4.85843156214280337125876407328, −4.41556530232110340639549623624, −4.19376897426909795331306892927, −3.02039232200028989237216684886, −2.51931836056213912077704320721, −1.84917066903373993670111953131, −1.78629603985942833701565101651, −0.55933767860152606293077920348,
0.55933767860152606293077920348, 1.78629603985942833701565101651, 1.84917066903373993670111953131, 2.51931836056213912077704320721, 3.02039232200028989237216684886, 4.19376897426909795331306892927, 4.41556530232110340639549623624, 4.85843156214280337125876407328, 5.34512157692928148289493860844, 5.65101637436830694972670324613, 6.10605778095352686373868321810, 7.02843128855991189148146013298, 7.15283522626009183354491701957, 7.54875238981829342683413107520, 8.025891254817114467795355804191, 8.453361736168553968633588603822, 8.605343572459844553586559349887, 9.285166336245128472599885348987, 9.694716791708732771998254663060, 9.730961028036697187419813221672
