| L(s) = 1 | + 3-s − 2·9-s + 4·11-s − 2·13-s + 6·17-s − 6·19-s + 5·23-s − 5·27-s + 9·29-s + 2·31-s + 4·33-s + 4·37-s − 2·39-s − 3·41-s + 43-s + 8·47-s + 6·51-s + 14·53-s − 6·57-s − 6·59-s + 61-s + 5·67-s + 5·69-s − 14·71-s + 2·73-s − 10·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.37·19-s + 1.04·23-s − 0.962·27-s + 1.67·29-s + 0.359·31-s + 0.696·33-s + 0.657·37-s − 0.320·39-s − 0.468·41-s + 0.152·43-s + 1.16·47-s + 0.840·51-s + 1.92·53-s − 0.794·57-s − 0.781·59-s + 0.128·61-s + 0.610·67-s + 0.601·69-s − 1.66·71-s + 0.234·73-s − 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.532123555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.532123555\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.13566883031346, −13.63235581500974, −13.08189376546858, −12.42921382698493, −12.02409644713348, −11.68509347834959, −11.05354030777888, −10.32111578325519, −10.11386417551765, −9.352306728180701, −8.856651370511591, −8.585750263435070, −8.002099937782250, −7.346284988387714, −6.924083102833636, −6.139110194203900, −5.911147974862815, −5.042361360290806, −4.517566220023659, −3.881076182348108, −3.287054231369577, −2.710152323512211, −2.180573021622963, −1.239414717205133, −0.6412144757779780,
0.6412144757779780, 1.239414717205133, 2.180573021622963, 2.710152323512211, 3.287054231369577, 3.881076182348108, 4.517566220023659, 5.042361360290806, 5.911147974862815, 6.139110194203900, 6.924083102833636, 7.346284988387714, 8.002099937782250, 8.585750263435070, 8.856651370511591, 9.352306728180701, 10.11386417551765, 10.32111578325519, 11.05354030777888, 11.68509347834959, 12.02409644713348, 12.42921382698493, 13.08189376546858, 13.63235581500974, 14.13566883031346
