| L(s) = 1 | + 8·7-s − 9-s − 12·17-s + 8·23-s − 25-s − 16·31-s + 12·41-s − 24·47-s + 34·49-s − 8·63-s − 16·71-s − 20·73-s + 32·79-s + 81-s + 12·89-s + 36·97-s − 8·103-s − 12·113-s − 96·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 1/3·9-s − 2.91·17-s + 1.66·23-s − 1/5·25-s − 2.87·31-s + 1.87·41-s − 3.50·47-s + 34/7·49-s − 1.00·63-s − 1.89·71-s − 2.34·73-s + 3.60·79-s + 1/9·81-s + 1.27·89-s + 3.65·97-s − 0.788·103-s − 1.12·113-s − 8.80·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.658623577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.658623577\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.818304859281466886705459347618, −8.114509441961146371820599241346, −8.107313899618519193243333192694, −7.60995039934569639130895055044, −7.31539773881227038269774305709, −6.87471635599260470475732848724, −6.60978671054169850082020094874, −5.91751928507150151242326553547, −5.69184154217550836471767929470, −5.13189839497178863615499394561, −4.71342589549645101201973542954, −4.60635796842598833397739656625, −4.49058188743371405713228700550, −3.58603221115160160089037658277, −3.36150848214147863793051141359, −2.33971120010145294182726859837, −2.27005751555819795067440274749, −1.68120931542836295490257074975, −1.44280350857290296834054919541, −0.45670117365139140563829090165,
0.45670117365139140563829090165, 1.44280350857290296834054919541, 1.68120931542836295490257074975, 2.27005751555819795067440274749, 2.33971120010145294182726859837, 3.36150848214147863793051141359, 3.58603221115160160089037658277, 4.49058188743371405713228700550, 4.60635796842598833397739656625, 4.71342589549645101201973542954, 5.13189839497178863615499394561, 5.69184154217550836471767929470, 5.91751928507150151242326553547, 6.60978671054169850082020094874, 6.87471635599260470475732848724, 7.31539773881227038269774305709, 7.60995039934569639130895055044, 8.107313899618519193243333192694, 8.114509441961146371820599241346, 8.818304859281466886705459347618
