| L(s) = 1 | + 1.44·3-s − 0.898·9-s + 0.550·11-s − 7.89·17-s + 8.34·19-s − 5.65·27-s + 0.797·33-s − 12.7·41-s − 10·43-s − 7·49-s − 11.4·51-s + 12.1·57-s − 6·59-s + 14.3·67-s + 13.6·73-s − 5.49·81-s − 11.4·83-s + 13.8·89-s − 10·97-s − 0.494·99-s − 20.1·107-s − 0.797·113-s + ⋯ |
| L(s) = 1 | + 0.836·3-s − 0.299·9-s + 0.165·11-s − 1.91·17-s + 1.91·19-s − 1.08·27-s + 0.138·33-s − 1.99·41-s − 1.52·43-s − 49-s − 1.60·51-s + 1.60·57-s − 0.781·59-s + 1.75·67-s + 1.60·73-s − 0.610·81-s − 1.25·83-s + 1.47·89-s − 1.01·97-s − 0.0497·99-s − 1.94·107-s − 0.0750·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.44T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 0.550T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.89T + 17T^{2} \) |
| 19 | \( 1 - 8.34T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.88060018305853463525442801332, −6.90485898506978115661051188541, −6.50662421180971289549145324764, −5.37644926878243155036929436780, −4.86340413548539080120202098662, −3.78442074354798081244950755146, −3.20671257835573326590524310896, −2.38742513462064796930137608277, −1.51453439629908101213765663735, 0,
1.51453439629908101213765663735, 2.38742513462064796930137608277, 3.20671257835573326590524310896, 3.78442074354798081244950755146, 4.86340413548539080120202098662, 5.37644926878243155036929436780, 6.50662421180971289549145324764, 6.90485898506978115661051188541, 7.88060018305853463525442801332
