| L(s) = 1 | + 0.568·3-s − 2.67·9-s + 5.17·11-s + 3.51·13-s + 2.67·17-s − 8.14·19-s + 3.87·23-s − 3.22·27-s + 1.49·29-s + 0.447·31-s + 2.93·33-s − 10.4·37-s + 2·39-s + 6.46·41-s + 8.45·43-s + 6.07·47-s + 1.51·51-s + 7.36·53-s − 4.63·57-s + 0.544·59-s − 11.8·61-s − 6.60·67-s + 2.20·69-s + 3.10·71-s + 1.02·73-s + 9.08·79-s + 6.19·81-s + ⋯ |
| L(s) = 1 | + 0.328·3-s − 0.892·9-s + 1.55·11-s + 0.975·13-s + 0.648·17-s − 1.86·19-s + 0.807·23-s − 0.621·27-s + 0.277·29-s + 0.0803·31-s + 0.511·33-s − 1.71·37-s + 0.320·39-s + 1.01·41-s + 1.28·43-s + 0.886·47-s + 0.212·51-s + 1.01·53-s − 0.613·57-s + 0.0709·59-s − 1.51·61-s − 0.807·67-s + 0.265·69-s + 0.368·71-s + 0.120·73-s + 1.02·79-s + 0.688·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.505446444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.505446444\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.568T + 3T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 13 | \( 1 - 3.51T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 + 8.14T + 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 - 1.49T + 29T^{2} \) |
| 31 | \( 1 - 0.447T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 6.46T + 41T^{2} \) |
| 43 | \( 1 - 8.45T + 43T^{2} \) |
| 47 | \( 1 - 6.07T + 47T^{2} \) |
| 53 | \( 1 - 7.36T + 53T^{2} \) |
| 59 | \( 1 - 0.544T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 - 3.10T + 71T^{2} \) |
| 73 | \( 1 - 1.02T + 73T^{2} \) |
| 79 | \( 1 - 9.08T + 79T^{2} \) |
| 83 | \( 1 + 7.18T + 83T^{2} \) |
| 89 | \( 1 - 5.60T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.70043346969830663886035668723, −6.94790766598510759834395389584, −6.16006991041605872631826768723, −5.92778907308272635132277004497, −4.84188234611486422133468531513, −3.98485185518557869388014034914, −3.55216425857004683533507807205, −2.62361818873995847681930136694, −1.72283341810739283731252551178, −0.76045461591554717823154627777,
0.76045461591554717823154627777, 1.72283341810739283731252551178, 2.62361818873995847681930136694, 3.55216425857004683533507807205, 3.98485185518557869388014034914, 4.84188234611486422133468531513, 5.92778907308272635132277004497, 6.16006991041605872631826768723, 6.94790766598510759834395389584, 7.70043346969830663886035668723
