| L(s) = 1 | − 1.02e3·2-s + 1.04e6·4-s − 1.07e9·8-s + 4.23e10·11-s + 1.09e12·16-s + 3.35e12·17-s − 1.01e12·19-s − 4.34e13·22-s + 9.53e13·25-s − 1.12e15·32-s − 3.43e15·34-s + 1.03e15·38-s + 2.54e16·41-s + 2.78e15·43-s + 4.44e16·44-s + 7.97e16·49-s − 9.76e16·50-s + 1.73e17·59-s + 1.15e18·64-s − 3.56e17·67-s + 3.51e18·68-s − 6.01e18·73-s − 1.06e18·76-s − 2.60e19·82-s + 3.10e19·83-s − 2.84e18·86-s − 4.55e19·88-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 1.63·11-s + 16-s + 1.66·17-s − 0.165·19-s − 1.63·22-s + 25-s − 32-s − 1.66·34-s + 0.165·38-s + 1.89·41-s + 0.128·43-s + 1.63·44-s + 49-s − 50-s + 0.340·59-s + 64-s − 0.195·67-s + 1.66·68-s − 1.40·73-s − 0.165·76-s − 1.89·82-s + 1.99·83-s − 0.128·86-s − 1.63·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(1.967758104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.967758104\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{10} T \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 7 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 11 | \( 1 - 42383023726 T + p^{20} T^{2} \) |
| 13 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 17 | \( 1 - 3353535763774 T + p^{20} T^{2} \) |
| 19 | \( 1 + 1014654432526 T + p^{20} T^{2} \) |
| 23 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 29 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 31 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 37 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 41 | \( 1 - 25418071370591326 T + p^{20} T^{2} \) |
| 43 | \( 1 - 2781113986388498 T + p^{20} T^{2} \) |
| 47 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 53 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 59 | \( 1 - 173912197184497198 T + p^{20} T^{2} \) |
| 61 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 67 | \( 1 + 356137514166464974 T + p^{20} T^{2} \) |
| 71 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 73 | \( 1 + 6016717170316692574 T + p^{20} T^{2} \) |
| 79 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 83 | \( 1 - 31022856480301602574 T + p^{20} T^{2} \) |
| 89 | \( 1 + 61202446863210984674 T + p^{20} T^{2} \) |
| 97 | \( 1 + 50009130514058267902 T + p^{20} T^{2} \) |
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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
−10.69101734496217098666848402140, −9.617155241152997180504483296189, −8.845081649352637385181319401699, −7.68334532172466686784337896805, −6.69234139085407194074509950030, −5.67647835646137525879441359004, −3.95579651647993289169161350965, −2.81385343414303355035035655005, −1.42926763420696392447469492068, −0.78378243989395013061315096680,
0.78378243989395013061315096680, 1.42926763420696392447469492068, 2.81385343414303355035035655005, 3.95579651647993289169161350965, 5.67647835646137525879441359004, 6.69234139085407194074509950030, 7.68334532172466686784337896805, 8.845081649352637385181319401699, 9.617155241152997180504483296189, 10.69101734496217098666848402140
