| L(s) = 1 | − 1.23·5-s + 7-s + 2·11-s − 4.47·13-s + 1.23·17-s − 19-s − 2·23-s − 3.47·25-s − 5.70·29-s − 1.52·31-s − 1.23·35-s + 8.47·37-s + 8.47·41-s + 8·43-s − 1.70·47-s + 49-s − 11.2·53-s − 2.47·55-s + 8.94·59-s − 13.4·61-s + 5.52·65-s − 1.52·67-s − 7.70·71-s + 12.4·73-s + 2·77-s + 4·79-s + 13.7·83-s + ⋯ |
| L(s) = 1 | − 0.552·5-s + 0.377·7-s + 0.603·11-s − 1.24·13-s + 0.299·17-s − 0.229·19-s − 0.417·23-s − 0.694·25-s − 1.05·29-s − 0.274·31-s − 0.208·35-s + 1.39·37-s + 1.32·41-s + 1.21·43-s − 0.249·47-s + 0.142·49-s − 1.54·53-s − 0.333·55-s + 1.16·59-s − 1.71·61-s + 0.685·65-s − 0.186·67-s − 0.914·71-s + 1.45·73-s + 0.227·77-s + 0.450·79-s + 1.50·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.500740555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500740555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.67522543234814574563635983307, −7.25019596556098432038560281367, −6.24525672208393597435706760113, −5.72162521374634430178265625180, −4.77012150595774402397662199076, −4.25135340697210115009910119665, −3.54231726191946630869870498148, −2.55154079721994403318654530298, −1.78316418100372479469732440228, −0.57669994224633301783839881854,
0.57669994224633301783839881854, 1.78316418100372479469732440228, 2.55154079721994403318654530298, 3.54231726191946630869870498148, 4.25135340697210115009910119665, 4.77012150595774402397662199076, 5.72162521374634430178265625180, 6.24525672208393597435706760113, 7.25019596556098432038560281367, 7.67522543234814574563635983307
