| L(s) = 1 | − 1.31e5·2-s + 1.25e8·3-s + 1.71e10·4-s − 1.64e13·6-s − 2.25e15·8-s − 1.00e15·9-s − 7.52e17·11-s + 2.15e18·12-s + 2.95e20·16-s + 1.39e21·17-s + 1.32e20·18-s + 1.78e20·19-s + 9.86e22·22-s − 2.81e23·24-s + 5.82e23·25-s − 2.21e24·27-s − 3.86e25·32-s − 9.42e25·33-s − 1.82e26·34-s − 1.73e25·36-s − 2.33e25·38-s + 3.36e27·41-s + 4.15e27·43-s − 1.29e28·44-s + 3.69e28·48-s + 5.41e28·49-s − 7.62e28·50-s + ⋯ |
| L(s) = 1 | − 2-s + 0.969·3-s + 4-s − 0.969·6-s − 8-s − 0.0604·9-s − 1.48·11-s + 0.969·12-s + 16-s + 1.68·17-s + 0.0604·18-s + 0.0324·19-s + 1.48·22-s − 0.969·24-s + 25-s − 1.02·27-s − 32-s − 1.44·33-s − 1.68·34-s − 0.0604·36-s − 0.0324·38-s + 1.28·41-s + 0.707·43-s − 1.48·44-s + 0.969·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{35}{2})\) |
\(\approx\) |
\(1.667113352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.667113352\) |
| \(L(18)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{17} T \) |
| good | 3 | \( 1 - 125174398 T + p^{34} T^{2} \) |
| 5 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 7 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 11 | \( 1 + 752887268851320946 T + p^{34} T^{2} \) |
| 13 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 17 | \( 1 - \)\(13\!\cdots\!82\)\( T + p^{34} T^{2} \) |
| 19 | \( 1 - \)\(17\!\cdots\!66\)\( T + p^{34} T^{2} \) |
| 23 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 29 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 31 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 37 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 41 | \( 1 - \)\(33\!\cdots\!54\)\( T + p^{34} T^{2} \) |
| 43 | \( 1 - \)\(41\!\cdots\!14\)\( T + p^{34} T^{2} \) |
| 47 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 53 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 59 | \( 1 + \)\(12\!\cdots\!62\)\( T + p^{34} T^{2} \) |
| 61 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 67 | \( 1 - \)\(20\!\cdots\!62\)\( T + p^{34} T^{2} \) |
| 71 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 73 | \( 1 - \)\(62\!\cdots\!58\)\( T + p^{34} T^{2} \) |
| 79 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 83 | \( 1 - \)\(46\!\cdots\!98\)\( T + p^{34} T^{2} \) |
| 89 | \( 1 + \)\(16\!\cdots\!74\)\( T + p^{34} T^{2} \) |
| 97 | \( 1 + \)\(87\!\cdots\!74\)\( T + p^{34} T^{2} \) |
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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
−14.25420178953483050004267169671, −12.54186969267847206080553027433, −10.80914256147746859460849234925, −9.595672367797062934203320400947, −8.300232313604602375483952094346, −7.49828885233105467931267041738, −5.59124182288568018880272805853, −3.21722003070601227199721650162, −2.34153368080688005170263545904, −0.76299363815156898327234049595,
0.76299363815156898327234049595, 2.34153368080688005170263545904, 3.21722003070601227199721650162, 5.59124182288568018880272805853, 7.49828885233105467931267041738, 8.300232313604602375483952094346, 9.595672367797062934203320400947, 10.80914256147746859460849234925, 12.54186969267847206080553027433, 14.25420178953483050004267169671
