| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 0.645·11-s − 4.64·13-s + 16-s − 7.29·17-s + 6.29·19-s − 20-s − 0.645·22-s + 3·23-s + 25-s − 4.64·26-s + 2·31-s + 32-s − 7.29·34-s + 9.93·37-s + 6.29·38-s − 40-s + 6.64·41-s + 0.708·43-s − 0.645·44-s + 3·46-s + 11.5·47-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.194·11-s − 1.28·13-s + 0.250·16-s − 1.76·17-s + 1.44·19-s − 0.223·20-s − 0.137·22-s + 0.625·23-s + 0.200·25-s − 0.911·26-s + 0.359·31-s + 0.176·32-s − 1.25·34-s + 1.63·37-s + 1.02·38-s − 0.158·40-s + 1.03·41-s + 0.108·43-s − 0.0973·44-s + 0.442·46-s + 1.68·47-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.513448351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.513448351\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 0.645T + 11T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 9.93T + 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 - 0.708T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 1.29T + 59T^{2} \) |
| 61 | \( 1 - 0.708T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 0.708T + 79T^{2} \) |
| 83 | \( 1 - 4.70T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.224612407374561864459874219950, −7.29225837033198205962304340408, −7.12825745597651647194803175703, −6.07146594367169852092749559929, −5.27765005433949840976188675200, −4.57389320587775153250067266461, −3.99269470374338941499724672115, −2.81362293862829108362201136848, −2.34586965416626740885020585846, −0.78465737339981128052290300713,
0.78465737339981128052290300713, 2.34586965416626740885020585846, 2.81362293862829108362201136848, 3.99269470374338941499724672115, 4.57389320587775153250067266461, 5.27765005433949840976188675200, 6.07146594367169852092749559929, 7.12825745597651647194803175703, 7.29225837033198205962304340408, 8.224612407374561864459874219950
