| L(s) = 1 | − 71·7-s − 337·13-s + 601·19-s + 625·25-s − 194·31-s − 529·37-s + 3.21e3·43-s + 2.64e3·49-s + 7.19e3·61-s − 2.90e3·67-s − 1.24e3·73-s − 4.67e3·79-s + 2.39e4·91-s + 9.07e3·97-s + 1.98e4·103-s + 2.20e4·109-s + ⋯ |
| L(s) = 1 | − 1.44·7-s − 1.99·13-s + 1.66·19-s + 25-s − 0.201·31-s − 0.386·37-s + 1.73·43-s + 1.09·49-s + 1.93·61-s − 0.646·67-s − 0.234·73-s − 0.749·79-s + 2.88·91-s + 0.964·97-s + 1.87·103-s + 1.85·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.264584993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264584993\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( 1 + 71 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 + 337 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 - 601 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 + 194 T + p^{4} T^{2} \) |
| 37 | \( 1 + 529 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 3214 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 7199 T + p^{4} T^{2} \) |
| 67 | \( 1 + 2903 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 1249 T + p^{4} T^{2} \) |
| 79 | \( 1 + 4679 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 - 9071 T + p^{4} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.24108862921957349297748381128, −9.713268471192807607968872442653, −8.978250110956210275645706894229, −7.47811438929592644837743066921, −6.99246271381347702919650458999, −5.77714356638782193393030115398, −4.79169773321565554220310696048, −3.37452198178609414678204111534, −2.51370563586912541520648740333, −0.61670556832565175506583207071,
0.61670556832565175506583207071, 2.51370563586912541520648740333, 3.37452198178609414678204111534, 4.79169773321565554220310696048, 5.77714356638782193393030115398, 6.99246271381347702919650458999, 7.47811438929592644837743066921, 8.978250110956210275645706894229, 9.713268471192807607968872442653, 10.24108862921957349297748381128
