| L(s) = 1 | − 0.799·3-s + 0.913·5-s − 2.36·9-s − 1.57·11-s − 5.79·13-s − 0.731·15-s + 0.695·17-s − 7.46·19-s + 3.49·23-s − 4.16·25-s + 4.28·27-s + 8.00·29-s + 10.5·31-s + 1.25·33-s + 0.654·37-s + 4.63·39-s + 11.2·41-s + 8.88·43-s − 2.15·45-s + 5.15·47-s − 0.556·51-s + 4.88·53-s − 1.43·55-s + 5.97·57-s − 5.84·59-s − 10.3·61-s − 5.29·65-s + ⋯ |
| L(s) = 1 | − 0.461·3-s + 0.408·5-s − 0.786·9-s − 0.474·11-s − 1.60·13-s − 0.188·15-s + 0.168·17-s − 1.71·19-s + 0.729·23-s − 0.832·25-s + 0.825·27-s + 1.48·29-s + 1.90·31-s + 0.218·33-s + 0.107·37-s + 0.741·39-s + 1.76·41-s + 1.35·43-s − 0.321·45-s + 0.752·47-s − 0.0779·51-s + 0.671·53-s − 0.193·55-s + 0.791·57-s − 0.760·59-s − 1.32·61-s − 0.656·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.134200591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.134200591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.799T + 3T^{2} \) |
| 5 | \( 1 - 0.913T + 5T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 13 | \( 1 + 5.79T + 13T^{2} \) |
| 17 | \( 1 - 0.695T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 - 3.49T + 23T^{2} \) |
| 29 | \( 1 - 8.00T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 0.654T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 8.88T + 43T^{2} \) |
| 47 | \( 1 - 5.15T + 47T^{2} \) |
| 53 | \( 1 - 4.88T + 53T^{2} \) |
| 59 | \( 1 + 5.84T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 3.06T + 67T^{2} \) |
| 71 | \( 1 - 9.60T + 71T^{2} \) |
| 73 | \( 1 + 9.14T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 4.19T + 83T^{2} \) |
| 89 | \( 1 + 7.84T + 89T^{2} \) |
| 97 | \( 1 + 7.55T + 97T^{2} \) |
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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
−8.810104649113568307736575218318, −8.074931279498194301579410396436, −7.30395619892580490130353313790, −6.32608280658938114565866691469, −5.87018149655534329763467298203, −4.87278858137525716736122939817, −4.36647997253124289898540030771, −2.75869186612610958812693783096, −2.38644551908933902972919501131, −0.64686744592065137547737595743,
0.64686744592065137547737595743, 2.38644551908933902972919501131, 2.75869186612610958812693783096, 4.36647997253124289898540030771, 4.87278858137525716736122939817, 5.87018149655534329763467298203, 6.32608280658938114565866691469, 7.30395619892580490130353313790, 8.074931279498194301579410396436, 8.810104649113568307736575218318
