| L(s) = 1 | − 27·9-s + 92·13-s − 104·17-s + 130·29-s − 396·37-s + 230·41-s − 343·49-s − 572·53-s − 830·61-s − 592·73-s + 729·81-s + 1.67e3·89-s − 1.81e3·97-s + 598·101-s − 1.74e3·109-s − 1.32e3·113-s − 2.48e3·117-s + ⋯ |
| L(s) = 1 | − 9-s + 1.96·13-s − 1.48·17-s + 0.832·29-s − 1.75·37-s + 0.876·41-s − 49-s − 1.48·53-s − 1.74·61-s − 0.949·73-s + 81-s + 1.98·89-s − 1.90·97-s + 0.589·101-s − 1.53·109-s − 1.10·113-s − 1.96·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 92 T + p^{3} T^{2} \) |
| 17 | \( 1 + 104 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + 396 T + p^{3} T^{2} \) |
| 41 | \( 1 - 230 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + 572 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 830 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 592 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 - 1670 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1816 T + p^{3} T^{2} \) |
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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
−9.157089213902084815610921872500, −8.705653971261654931611904565034, −7.934516173261182572230118215992, −6.57225643444512645066283619176, −6.10670708694178243260629000415, −4.96476862473451641774937930090, −3.84803395814460743200556269082, −2.86179031973489448577720907630, −1.50626567755976094592247748621, 0,
1.50626567755976094592247748621, 2.86179031973489448577720907630, 3.84803395814460743200556269082, 4.96476862473451641774937930090, 6.10670708694178243260629000415, 6.57225643444512645066283619176, 7.934516173261182572230118215992, 8.705653971261654931611904565034, 9.157089213902084815610921872500
