| L(s) = 1 | − 2.54·3-s − 0.780·7-s + 3.45·9-s + 5.16·11-s − 3.34·13-s + 6.62·17-s + 6.40·19-s + 1.98·21-s − 23-s − 1.15·27-s − 0.0877·29-s + 3.29·31-s − 13.1·33-s + 4.68·37-s + 8.50·39-s − 5.75·41-s − 2.62·43-s − 5.13·47-s − 6.39·49-s − 16.8·51-s − 8.80·53-s − 16.2·57-s + 2.98·59-s − 2.05·61-s − 2.69·63-s + 3.60·67-s + 2.54·69-s + ⋯ |
| L(s) = 1 | − 1.46·3-s − 0.295·7-s + 1.15·9-s + 1.55·11-s − 0.928·13-s + 1.60·17-s + 1.47·19-s + 0.432·21-s − 0.208·23-s − 0.223·27-s − 0.0162·29-s + 0.591·31-s − 2.28·33-s + 0.769·37-s + 1.36·39-s − 0.899·41-s − 0.400·43-s − 0.748·47-s − 0.912·49-s − 2.35·51-s − 1.20·53-s − 2.15·57-s + 0.388·59-s − 0.263·61-s − 0.339·63-s + 0.440·67-s + 0.305·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.198424929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198424929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.54T + 3T^{2} \) |
| 7 | \( 1 + 0.780T + 7T^{2} \) |
| 11 | \( 1 - 5.16T + 11T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 17 | \( 1 - 6.62T + 17T^{2} \) |
| 19 | \( 1 - 6.40T + 19T^{2} \) |
| 29 | \( 1 + 0.0877T + 29T^{2} \) |
| 31 | \( 1 - 3.29T + 31T^{2} \) |
| 37 | \( 1 - 4.68T + 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 47 | \( 1 + 5.13T + 47T^{2} \) |
| 53 | \( 1 + 8.80T + 53T^{2} \) |
| 59 | \( 1 - 2.98T + 59T^{2} \) |
| 61 | \( 1 + 2.05T + 61T^{2} \) |
| 67 | \( 1 - 3.60T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 3.43T + 79T^{2} \) |
| 83 | \( 1 + 3.62T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 0.427T + 97T^{2} \) |
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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
−8.129457702731393416396594879117, −7.45552843160800443399563019391, −6.58875436385154915911588270858, −6.26952816259784275514031713392, −5.27446112674846903234980429007, −4.94812252217822525324622749802, −3.83312049319569116926907133026, −3.07226877912482856153884325216, −1.52089358559558806857771082624, −0.71483432997993236563125489025,
0.71483432997993236563125489025, 1.52089358559558806857771082624, 3.07226877912482856153884325216, 3.83312049319569116926907133026, 4.94812252217822525324622749802, 5.27446112674846903234980429007, 6.26952816259784275514031713392, 6.58875436385154915911588270858, 7.45552843160800443399563019391, 8.129457702731393416396594879117
