| L(s) = 1 | − 2.37·3-s + 5-s − 7-s + 2.62·9-s − 6.37·11-s − 4.37·13-s − 2.37·15-s − 0.372·17-s + 4.74·19-s + 2.37·21-s − 4.74·23-s + 25-s + 0.883·27-s + 4.37·29-s − 8·31-s + 15.1·33-s − 35-s + 2·37-s + 10.3·39-s + 6.74·41-s + 8.74·43-s + 2.62·45-s − 7.11·47-s + 49-s + 0.883·51-s − 10.7·53-s − 6.37·55-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 0.447·5-s − 0.377·7-s + 0.875·9-s − 1.92·11-s − 1.21·13-s − 0.612·15-s − 0.0902·17-s + 1.08·19-s + 0.517·21-s − 0.989·23-s + 0.200·25-s + 0.169·27-s + 0.811·29-s − 1.43·31-s + 2.63·33-s − 0.169·35-s + 0.328·37-s + 1.66·39-s + 1.05·41-s + 1.33·43-s + 0.391·45-s − 1.03·47-s + 0.142·49-s + 0.123·51-s − 1.47·53-s − 0.859·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5765967361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5765967361\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 11 | \( 1 + 6.37T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 + 0.372T + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.74T + 41T^{2} \) |
| 43 | \( 1 - 8.74T + 43T^{2} \) |
| 47 | \( 1 + 7.11T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 2.74T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 9.86T + 97T^{2} \) |
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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
−9.395974814950879083279226335886, −7.962701631456653947542615130948, −7.49301554514809412248307777584, −6.53010976196831502227079732817, −5.72820970159234419898133277402, −5.23589701963254618285651913701, −4.59467857619858477638165897048, −3.08949764723252265928376375935, −2.17898061949279373872554101835, −0.49933168956732824256766810596,
0.49933168956732824256766810596, 2.17898061949279373872554101835, 3.08949764723252265928376375935, 4.59467857619858477638165897048, 5.23589701963254618285651913701, 5.72820970159234419898133277402, 6.53010976196831502227079732817, 7.49301554514809412248307777584, 7.962701631456653947542615130948, 9.395974814950879083279226335886
