| L(s) = 1 | + 6.55e4·2-s + 5.70e7·3-s + 4.29e9·4-s + 3.74e12·6-s + 2.81e14·8-s + 1.40e15·9-s + 3.75e15·11-s + 2.45e17·12-s + 1.84e19·16-s + 5.72e19·17-s + 9.21e19·18-s + 2.49e20·19-s + 2.45e20·22-s + 1.60e22·24-s + 2.32e22·25-s − 2.54e22·27-s + 1.20e24·32-s + 2.14e23·33-s + 3.75e24·34-s + 6.04e24·36-s + 1.63e25·38-s − 1.26e26·41-s − 2.36e26·43-s + 1.61e25·44-s + 1.05e27·48-s + 1.10e27·49-s + 1.52e27·50-s + ⋯ |
| L(s) = 1 | + 2-s + 1.32·3-s + 4-s + 1.32·6-s + 8-s + 0.759·9-s + 0.0816·11-s + 1.32·12-s + 16-s + 1.17·17-s + 0.759·18-s + 0.863·19-s + 0.0816·22-s + 1.32·24-s + 25-s − 0.319·27-s + 32-s + 0.108·33-s + 1.17·34-s + 0.759·36-s + 0.863·38-s − 1.98·41-s − 1.73·43-s + 0.0816·44-s + 1.32·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(7.123437892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.123437892\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{16} T \) |
| good | 3 | \( 1 - 57091714 T + p^{32} T^{2} \) |
| 5 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 7 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 11 | \( 1 - 3750819928010114 T + p^{32} T^{2} \) |
| 13 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 17 | \( 1 - 57264303392945459714 T + p^{32} T^{2} \) |
| 19 | \( 1 - \)\(24\!\cdots\!34\)\( T + p^{32} T^{2} \) |
| 23 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 29 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 31 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 37 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 41 | \( 1 + \)\(12\!\cdots\!46\)\( T + p^{32} T^{2} \) |
| 43 | \( 1 + \)\(23\!\cdots\!98\)\( T + p^{32} T^{2} \) |
| 47 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 53 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 59 | \( 1 - \)\(41\!\cdots\!82\)\( T + p^{32} T^{2} \) |
| 61 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 67 | \( 1 - \)\(51\!\cdots\!34\)\( T + p^{32} T^{2} \) |
| 71 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 73 | \( 1 + \)\(10\!\cdots\!06\)\( T + p^{32} T^{2} \) |
| 79 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 83 | \( 1 - \)\(27\!\cdots\!94\)\( T + p^{32} T^{2} \) |
| 89 | \( 1 + \)\(29\!\cdots\!46\)\( T + p^{32} T^{2} \) |
| 97 | \( 1 + \)\(29\!\cdots\!58\)\( T + p^{32} T^{2} \) |
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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
−14.27592842400826640561785646283, −13.28693527041105297313448455071, −11.84799085383215384315704267919, −10.02932354157754562046464944805, −8.346918114322761883070107526701, −7.07329669497693029954449383948, −5.28497410607252547266957829421, −3.65624379022894930503600224716, −2.81274892269836667526510958449, −1.43373109151404335645499216578,
1.43373109151404335645499216578, 2.81274892269836667526510958449, 3.65624379022894930503600224716, 5.28497410607252547266957829421, 7.07329669497693029954449383948, 8.346918114322761883070107526701, 10.02932354157754562046464944805, 11.84799085383215384315704267919, 13.28693527041105297313448455071, 14.27592842400826640561785646283
