| L(s) = 1 | + 9·3-s + 81·9-s − 337·13-s + 647·19-s + 625·25-s + 729·27-s + 1.55e3·31-s − 529·37-s − 3.03e3·39-s + 3.19e3·43-s + 5.82e3·57-s − 1.96e3·61-s + 2.90e3·67-s + 9.79e3·73-s + 5.62e3·75-s + 4.67e3·79-s + 6.56e3·81-s + 1.40e4·93-s − 1.88e4·97-s − 1.98e4·103-s − 3.31e3·109-s − 4.76e3·111-s − 2.72e4·117-s + ⋯ |
| L(s) = 1 | + 3-s + 9-s − 1.99·13-s + 1.79·19-s + 25-s + 27-s + 1.62·31-s − 0.386·37-s − 1.99·39-s + 1.72·43-s + 1.79·57-s − 0.528·61-s + 0.646·67-s + 1.83·73-s + 75-s + 0.749·79-s + 81-s + 1.62·93-s − 1.99·97-s − 1.87·103-s − 0.278·109-s − 0.386·111-s − 1.99·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.153063683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.153063683\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 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 | \( 1 - 647 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 - 1559 T + p^{4} T^{2} \) |
| 37 | \( 1 + 529 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 3191 T + p^{4} T^{2} \) |
| 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 | \( 1 + 1966 T + p^{4} T^{2} \) |
| 67 | \( 1 - 2903 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 - 9791 T + p^{4} T^{2} \) |
| 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 | \( 1 + 18814 T + p^{4} T^{2} \) |
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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
−9.722726848448231123391351553477, −9.470762997750212841661337433081, −8.262718632397893983436497393536, −7.48780649485234246472414282767, −6.83713731128977215392219372842, −5.27715324574866784583994626971, −4.45872537101706452406882209744, −3.12178107877366830501203788630, −2.39416710819565981317129057725, −0.924824435754361684809604442499,
0.924824435754361684809604442499, 2.39416710819565981317129057725, 3.12178107877366830501203788630, 4.45872537101706452406882209744, 5.27715324574866784583994626971, 6.83713731128977215392219372842, 7.48780649485234246472414282767, 8.262718632397893983436497393536, 9.470762997750212841661337433081, 9.722726848448231123391351553477
