| L(s) = 1 | + 3-s − 3.24·5-s + 3.21·7-s + 9-s − 2.82·11-s − 5.85·13-s − 3.24·15-s + 1.86·17-s − 5.37·19-s + 3.21·21-s − 4.21·23-s + 5.51·25-s + 27-s + 3.33·29-s − 1.27·31-s − 2.82·33-s − 10.4·35-s − 2.06·37-s − 5.85·39-s + 2.10·41-s − 6.96·43-s − 3.24·45-s + 6.68·47-s + 3.30·49-s + 1.86·51-s + 9.34·53-s + 9.14·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.45·5-s + 1.21·7-s + 0.333·9-s − 0.850·11-s − 1.62·13-s − 0.837·15-s + 0.451·17-s − 1.23·19-s + 0.700·21-s − 0.877·23-s + 1.10·25-s + 0.192·27-s + 0.618·29-s − 0.228·31-s − 0.491·33-s − 1.75·35-s − 0.339·37-s − 0.936·39-s + 0.329·41-s − 1.06·43-s − 0.483·45-s + 0.975·47-s + 0.472·49-s + 0.260·51-s + 1.28·53-s + 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.377183420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.377183420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 3.24T + 5T^{2} \) |
| 7 | \( 1 - 3.21T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 - 3.33T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 + 2.06T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 + 6.96T + 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 7.78T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 5.43T + 73T^{2} \) |
| 79 | \( 1 - 4.16T + 79T^{2} \) |
| 83 | \( 1 - 1.27T + 83T^{2} \) |
| 89 | \( 1 + 5.91T + 89T^{2} \) |
| 97 | \( 1 - 4.01T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.86504014965749548351628126193, −7.42470452117701920869547923411, −6.78340177209240987339556669188, −5.50201046302284606744685713953, −4.84932023640126625868226205244, −4.29871851027464694281872247639, −3.63040520621395524552473219733, −2.56718366595464354073511876970, −2.01355615517176531464333349934, −0.53994823639919775613584511930,
0.53994823639919775613584511930, 2.01355615517176531464333349934, 2.56718366595464354073511876970, 3.63040520621395524552473219733, 4.29871851027464694281872247639, 4.84932023640126625868226205244, 5.50201046302284606744685713953, 6.78340177209240987339556669188, 7.42470452117701920869547923411, 7.86504014965749548351628126193
