| L(s) = 1 | − 6·5-s − 2·13-s + 6·17-s + 17·25-s − 20·29-s + 6·37-s − 9·49-s − 8·53-s + 12·65-s + 28·73-s − 36·85-s + 20·89-s − 4·97-s + 24·101-s + 22·109-s + 28·113-s − 2·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 120·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 0.554·13-s + 1.45·17-s + 17/5·25-s − 3.71·29-s + 0.986·37-s − 9/7·49-s − 1.09·53-s + 1.48·65-s + 3.27·73-s − 3.90·85-s + 2.11·89-s − 0.406·97-s + 2.38·101-s + 2.10·109-s + 2.63·113-s − 0.181·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 9.96·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8833564126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8833564126\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−8.515245810440513603401246871378, −8.179973661375063929673139627961, −7.75259560555948641402045641659, −7.67874062083405697403242203637, −7.37186909254632687565611469841, −7.25744860291917568378418205894, −6.42621022266829406806621269248, −6.26172620424009590100880126171, −5.50627003545573358916860676169, −5.39207713387039701470231963984, −4.69922555728763204763305805093, −4.54966527534769299636854543770, −3.95102930778469607942271498610, −3.62155193860924093862598511061, −3.29805567501927992257675175863, −3.21486104860728289780686034821, −2.10768653893582006582456553436, −1.86810220297277951543526585012, −0.77836902662681759368104591465, −0.39758180144335461761380036596,
0.39758180144335461761380036596, 0.77836902662681759368104591465, 1.86810220297277951543526585012, 2.10768653893582006582456553436, 3.21486104860728289780686034821, 3.29805567501927992257675175863, 3.62155193860924093862598511061, 3.95102930778469607942271498610, 4.54966527534769299636854543770, 4.69922555728763204763305805093, 5.39207713387039701470231963984, 5.50627003545573358916860676169, 6.26172620424009590100880126171, 6.42621022266829406806621269248, 7.25744860291917568378418205894, 7.37186909254632687565611469841, 7.67874062083405697403242203637, 7.75259560555948641402045641659, 8.179973661375063929673139627961, 8.515245810440513603401246871378
