| L(s) = 1 | − 1.93·3-s + 1.24·7-s + 0.747·9-s − 0.513·11-s + 6.15·13-s − 4.51·17-s − 19-s − 2.41·21-s − 5.86·23-s + 4.36·27-s + 6.62·29-s + 6.41·31-s + 0.994·33-s − 1.40·37-s − 11.9·39-s + 10.6·41-s − 3.04·43-s − 1.99·47-s − 5.44·49-s + 8.74·51-s − 14.0·53-s + 1.93·57-s − 4.34·59-s + 10.7·61-s + 0.932·63-s − 9.89·67-s + 11.3·69-s + ⋯ |
| L(s) = 1 | − 1.11·3-s + 0.471·7-s + 0.249·9-s − 0.154·11-s + 1.70·13-s − 1.09·17-s − 0.229·19-s − 0.526·21-s − 1.22·23-s + 0.839·27-s + 1.23·29-s + 1.15·31-s + 0.173·33-s − 0.231·37-s − 1.90·39-s + 1.66·41-s − 0.464·43-s − 0.290·47-s − 0.777·49-s + 1.22·51-s − 1.93·53-s + 0.256·57-s − 0.565·59-s + 1.37·61-s + 0.117·63-s − 1.20·67-s + 1.36·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.187856709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187856709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 7 | \( 1 - 1.24T + 7T^{2} \) |
| 11 | \( 1 + 0.513T + 11T^{2} \) |
| 13 | \( 1 - 6.15T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 - 6.62T + 29T^{2} \) |
| 31 | \( 1 - 6.41T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 3.04T + 43T^{2} \) |
| 47 | \( 1 + 1.99T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 9.89T + 67T^{2} \) |
| 71 | \( 1 - 7.42T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 2.56T + 79T^{2} \) |
| 83 | \( 1 + 7.50T + 83T^{2} \) |
| 89 | \( 1 + 7.85T + 89T^{2} \) |
| 97 | \( 1 - 6.74T + 97T^{2} \) |
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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
−8.329969858292361094229887419795, −7.983451765466953643050475616939, −6.50284192652144108068409233795, −6.44479818527058019354710437964, −5.61983152234551653815331344414, −4.71292219796688939119103390979, −4.15706055029957661762160157046, −2.98833209566256860990705508822, −1.77939889325735359663450105121, −0.68198939034702020857083765837,
0.68198939034702020857083765837, 1.77939889325735359663450105121, 2.98833209566256860990705508822, 4.15706055029957661762160157046, 4.71292219796688939119103390979, 5.61983152234551653815331344414, 6.44479818527058019354710437964, 6.50284192652144108068409233795, 7.983451765466953643050475616939, 8.329969858292361094229887419795
