| L(s) = 1 | − 2.26·3-s + 5-s + 1.11·7-s + 2.11·9-s + 5.37·11-s − 2.26·15-s − 7.90·19-s − 2.52·21-s + 6.49·23-s + 25-s + 2·27-s + 3.63·29-s + 3.14·31-s − 12.1·33-s + 1.11·35-s + 7.40·37-s + 4.75·41-s + 4.78·43-s + 2.11·45-s − 6.16·47-s − 5.75·49-s + 0.292·53-s + 5.37·55-s + 17.8·57-s − 11.3·59-s − 5.34·61-s + 2.36·63-s + ⋯ |
| L(s) = 1 | − 1.30·3-s + 0.447·5-s + 0.421·7-s + 0.705·9-s + 1.62·11-s − 0.583·15-s − 1.81·19-s − 0.550·21-s + 1.35·23-s + 0.200·25-s + 0.384·27-s + 0.675·29-s + 0.565·31-s − 2.11·33-s + 0.188·35-s + 1.21·37-s + 0.742·41-s + 0.729·43-s + 0.315·45-s − 0.898·47-s − 0.822·49-s + 0.0401·53-s + 0.725·55-s + 2.36·57-s − 1.48·59-s − 0.684·61-s + 0.297·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.441534917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441534917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.90T + 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 - 3.63T + 29T^{2} \) |
| 31 | \( 1 - 3.14T + 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 - 4.75T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 - 0.292T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 3.43T + 71T^{2} \) |
| 73 | \( 1 - 4.59T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 7.40T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 3.76T + 97T^{2} \) |
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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
−8.809172689395273398705282382817, −7.80382528809538244941006990293, −6.69711875150671493787951276820, −6.37825050974394212102978535251, −5.79071218668798591761283333036, −4.60215433537636180782343378281, −4.45079067812722091913498550190, −3.01333361736635170455142801204, −1.72689212719307816468248807149, −0.806781870174950945219295232268,
0.806781870174950945219295232268, 1.72689212719307816468248807149, 3.01333361736635170455142801204, 4.45079067812722091913498550190, 4.60215433537636180782343378281, 5.79071218668798591761283333036, 6.37825050974394212102978535251, 6.69711875150671493787951276820, 7.80382528809538244941006990293, 8.809172689395273398705282382817
