| L(s) = 1 | + 6·5-s + 9·9-s − 10·13-s − 30·17-s + 11·25-s − 42·29-s + 70·37-s + 18·41-s + 54·45-s + 49·49-s − 90·53-s + 22·61-s − 60·65-s − 110·73-s + 81·81-s − 180·85-s − 78·89-s + 130·97-s + 198·101-s + 182·109-s − 30·113-s − 90·117-s + ⋯ |
| L(s) = 1 | + 6/5·5-s + 9-s − 0.769·13-s − 1.76·17-s + 0.439·25-s − 1.44·29-s + 1.89·37-s + 0.439·41-s + 6/5·45-s + 49-s − 1.69·53-s + 0.360·61-s − 0.923·65-s − 1.50·73-s + 81-s − 2.11·85-s − 0.876·89-s + 1.34·97-s + 1.96·101-s + 1.66·109-s − 0.265·113-s − 0.769·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.386539719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.386539719\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( 1 - 6 T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 10 T + p^{2} T^{2} \) |
| 17 | \( 1 + 30 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 42 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 70 T + p^{2} T^{2} \) |
| 41 | \( 1 - 18 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 90 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 78 T + p^{2} T^{2} \) |
| 97 | \( 1 - 130 T + p^{2} T^{2} \) |
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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
−14.64584286276008033366841738313, −13.40280133451002265757830694315, −12.80520234624778956534761841212, −11.18066146400379094826178853010, −9.955133637695046826597564300549, −9.158494034457666168414053159986, −7.35339310334956040245109959360, −6.09502123689733974938890020251, −4.53511279429985186094893991488, −2.12052951372109079990428685251,
2.12052951372109079990428685251, 4.53511279429985186094893991488, 6.09502123689733974938890020251, 7.35339310334956040245109959360, 9.158494034457666168414053159986, 9.955133637695046826597564300549, 11.18066146400379094826178853010, 12.80520234624778956534761841212, 13.40280133451002265757830694315, 14.64584286276008033366841738313
