| L(s) = 1 | + 16.9·5-s − 11·7-s + 16.9·11-s − 29·13-s − 50.9·17-s + 29·19-s − 84.8·23-s + 162.·25-s − 271.·29-s + 268·31-s − 186.·35-s − 83·37-s − 271.·41-s − 232·43-s + 390.·47-s − 222·49-s − 305.·53-s + 287.·55-s + 288.·59-s − 767·61-s − 492.·65-s − 511·67-s − 712.·71-s + 137·73-s − 186.·77-s + 475·79-s + 576.·83-s + ⋯ |
| L(s) = 1 | + 1.51·5-s − 0.593·7-s + 0.465·11-s − 0.618·13-s − 0.726·17-s + 0.350·19-s − 0.769·23-s + 1.30·25-s − 1.73·29-s + 1.55·31-s − 0.901·35-s − 0.368·37-s − 1.03·41-s − 0.822·43-s + 1.21·47-s − 0.647·49-s − 0.791·53-s + 0.706·55-s + 0.636·59-s − 1.60·61-s − 0.939·65-s − 0.931·67-s − 1.19·71-s + 0.219·73-s − 0.276·77-s + 0.676·79-s + 0.763·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 16.9T + 125T^{2} \) |
| 7 | \( 1 + 11T + 343T^{2} \) |
| 11 | \( 1 - 16.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 29T + 2.19e3T^{2} \) |
| 17 | \( 1 + 50.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 271.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 268T + 2.97e4T^{2} \) |
| 37 | \( 1 + 83T + 5.06e4T^{2} \) |
| 41 | \( 1 + 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 232T + 7.95e4T^{2} \) |
| 47 | \( 1 - 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 288.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 767T + 2.26e5T^{2} \) |
| 67 | \( 1 + 511T + 3.00e5T^{2} \) |
| 71 | \( 1 + 712.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 137T + 3.89e5T^{2} \) |
| 79 | \( 1 - 475T + 4.93e5T^{2} \) |
| 83 | \( 1 - 576.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 254.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 821T + 9.12e5T^{2} \) |
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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
−8.850653817262544557718079502411, −7.71686710957708071953491148882, −6.72761168050695249150531021140, −6.20068602986309769301281043790, −5.43277232808547565440162677532, −4.51316487574786960336377884408, −3.31511885013609624508949690123, −2.31022282421937381901443560984, −1.51632200946093968524226060127, 0,
1.51632200946093968524226060127, 2.31022282421937381901443560984, 3.31511885013609624508949690123, 4.51316487574786960336377884408, 5.43277232808547565440162677532, 6.20068602986309769301281043790, 6.72761168050695249150531021140, 7.71686710957708071953491148882, 8.850653817262544557718079502411
