| L(s) = 1 | − 3-s + 9-s − 2·25-s − 27-s + 2·49-s + 24·59-s + 4·73-s + 2·75-s + 81-s − 4·97-s + 24·107-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 2·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s − 24·177-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 2/5·25-s − 0.192·27-s + 2/7·49-s + 3.12·59-s + 0.468·73-s + 0.230·75-s + 1/9·81-s − 0.406·97-s + 2.32·107-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.164·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s − 1.80·177-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.488316936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.488316936\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−7.74583868048918705583304639638, −7.33560649555241503748326163018, −6.94619456843931352166212388515, −6.57179677506898843905951095728, −6.05498608802269895216031943976, −5.66361373942718114697827466912, −5.27621961429003929477989927519, −4.78753583841009561813128674075, −4.31758645133063163767393127751, −3.74735906366057132239953766421, −3.42426092710094298655901904646, −2.54942324961895857431729956902, −2.15664370559519446720374886328, −1.33133794633126798828582037396, −0.55893394423636907455054582349,
0.55893394423636907455054582349, 1.33133794633126798828582037396, 2.15664370559519446720374886328, 2.54942324961895857431729956902, 3.42426092710094298655901904646, 3.74735906366057132239953766421, 4.31758645133063163767393127751, 4.78753583841009561813128674075, 5.27621961429003929477989927519, 5.66361373942718114697827466912, 6.05498608802269895216031943976, 6.57179677506898843905951095728, 6.94619456843931352166212388515, 7.33560649555241503748326163018, 7.74583868048918705583304639638
