| L(s) = 1 | + 3-s + 6·5-s − 6·11-s + 6·15-s + 12·17-s + 7·19-s + 19·25-s − 27-s + 5·31-s − 6·33-s − 37-s − 6·47-s + 12·51-s − 36·55-s + 7·57-s + 3·67-s − 15·73-s + 19·75-s − 27·79-s − 81-s + 12·83-s + 72·85-s − 12·89-s + 5·93-s + 42·95-s + 6·101-s + 5·103-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 2.68·5-s − 1.80·11-s + 1.54·15-s + 2.91·17-s + 1.60·19-s + 19/5·25-s − 0.192·27-s + 0.898·31-s − 1.04·33-s − 0.164·37-s − 0.875·47-s + 1.68·51-s − 4.85·55-s + 0.927·57-s + 0.366·67-s − 1.75·73-s + 2.19·75-s − 3.03·79-s − 1/9·81-s + 1.31·83-s + 7.80·85-s − 1.27·89-s + 0.518·93-s + 4.30·95-s + 0.597·101-s + 0.492·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.275211328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.275211328\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 15 T + 148 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 27 T + 322 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.589861023893819613100470247146, −8.610063771704793510265520121777, −8.596217096894313749291645414358, −8.059017848650272104376300752467, −7.51950458366559846925203790521, −7.45270248208334282845296132190, −6.94580057428965093040600054681, −6.12624659979445301914435287482, −6.03887503580458203500929696991, −5.59130185663011581115358520756, −5.29784443685783746924684808090, −5.16048421364952841532338198117, −4.59037233899916931811551422897, −3.69405827170958899028036813564, −3.08654920406461366848321048503, −2.83963854634259913635955965485, −2.66333488361478990893524094200, −1.76012839490133467533120247168, −1.52654114791102215398435488607, −0.842798662745513736023000483593,
0.842798662745513736023000483593, 1.52654114791102215398435488607, 1.76012839490133467533120247168, 2.66333488361478990893524094200, 2.83963854634259913635955965485, 3.08654920406461366848321048503, 3.69405827170958899028036813564, 4.59037233899916931811551422897, 5.16048421364952841532338198117, 5.29784443685783746924684808090, 5.59130185663011581115358520756, 6.03887503580458203500929696991, 6.12624659979445301914435287482, 6.94580057428965093040600054681, 7.45270248208334282845296132190, 7.51950458366559846925203790521, 8.059017848650272104376300752467, 8.596217096894313749291645414358, 8.610063771704793510265520121777, 9.589861023893819613100470247146
