| L(s) = 1 | + 7-s − 16·19-s + 7·25-s − 12·29-s + 10·31-s + 8·37-s − 12·47-s − 6·49-s − 6·53-s − 24·59-s − 30·83-s − 32·103-s + 40·109-s − 24·113-s − 5·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 3.67·19-s + 7/5·25-s − 2.22·29-s + 1.79·31-s + 1.31·37-s − 1.75·47-s − 6/7·49-s − 0.824·53-s − 3.12·59-s − 3.29·83-s − 3.15·103-s + 3.83·109-s − 2.25·113-s − 0.454·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5298671680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5298671680\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
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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
−8.903266117358267562791217637225, −8.438209848893797957704546536977, −8.263261845729627526623236309300, −7.80719320203608736589171944391, −7.59129733737160161103192656522, −6.87456772062442379134848680355, −6.55698003161024981615174731036, −6.35346546022633633832504714052, −6.01327416704652530955086595439, −5.51530470794969172823236751208, −4.86876758626749927299003927798, −4.59042925351812226200300028385, −4.31867201960808820661074573095, −3.95050717694923147376805777503, −3.23941534116565580275448290458, −2.76508530729559748218254686301, −2.40549431616391595933096612878, −1.59928495155205625869217911580, −1.51640728743610610827217440154, −0.21782973325106660387588493676,
0.21782973325106660387588493676, 1.51640728743610610827217440154, 1.59928495155205625869217911580, 2.40549431616391595933096612878, 2.76508530729559748218254686301, 3.23941534116565580275448290458, 3.95050717694923147376805777503, 4.31867201960808820661074573095, 4.59042925351812226200300028385, 4.86876758626749927299003927798, 5.51530470794969172823236751208, 6.01327416704652530955086595439, 6.35346546022633633832504714052, 6.55698003161024981615174731036, 6.87456772062442379134848680355, 7.59129733737160161103192656522, 7.80719320203608736589171944391, 8.263261845729627526623236309300, 8.438209848893797957704546536977, 8.903266117358267562791217637225
