| L(s) = 1 | + 1.17·3-s − 1.62·9-s + 6.62·11-s + 5.64·13-s + 7.99·17-s − 5.42·27-s − 0.623·29-s + 7.77·33-s + 6.62·39-s − 12.4·47-s + 9.37·51-s − 12·71-s − 13.4·73-s + 15.8·79-s − 1.49·81-s + 8.94·83-s − 0.731·87-s + 12.6·97-s − 10.7·99-s − 7.77·103-s − 9.87·109-s − 9.16·117-s + ⋯ |
| L(s) = 1 | + 0.677·3-s − 0.541·9-s + 1.99·11-s + 1.56·13-s + 1.93·17-s − 1.04·27-s − 0.115·29-s + 1.35·33-s + 1.06·39-s − 1.81·47-s + 1.31·51-s − 1.42·71-s − 1.57·73-s + 1.78·79-s − 0.165·81-s + 0.981·83-s − 0.0784·87-s + 1.28·97-s − 1.08·99-s − 0.765·103-s − 0.945·109-s − 0.847·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.179585983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.179585983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 11 | \( 1 - 6.62T + 11T^{2} \) |
| 13 | \( 1 - 5.64T + 13T^{2} \) |
| 17 | \( 1 - 7.99T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 0.623T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.289378356233317400043836541528, −7.75885708595038635420233078683, −6.74035074346664964922726509316, −6.11890010602130542888967617806, −5.52459314424103765757153694479, −4.31281518764731600420055245092, −3.46284291671323414524726911125, −3.25806179496169161883615951599, −1.75447112156164549141202643939, −1.05584080599531582497237232883,
1.05584080599531582497237232883, 1.75447112156164549141202643939, 3.25806179496169161883615951599, 3.46284291671323414524726911125, 4.31281518764731600420055245092, 5.52459314424103765757153694479, 6.11890010602130542888967617806, 6.74035074346664964922726509316, 7.75885708595038635420233078683, 8.289378356233317400043836541528
