| 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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.441314758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441314758\) |
| \(L(3)\) |
|
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
−10.57322551692122346605140395256, −9.149133745211011478281694415028, −8.455241400939526654755450728807, −7.21048149360478680903891967822, −6.31302846206040414118674109163, −5.71235740074028927498855028102, −4.48498921839492584711743042879, −3.62379386433542487577327766219, −1.85840745215919536365334667576, −0.69113241866691124296224348892,
0.69113241866691124296224348892, 1.85840745215919536365334667576, 3.62379386433542487577327766219, 4.48498921839492584711743042879, 5.71235740074028927498855028102, 6.31302846206040414118674109163, 7.21048149360478680903891967822, 8.455241400939526654755450728807, 9.149133745211011478281694415028, 10.57322551692122346605140395256
