| L(s) = 1 | − 2.75·3-s + 1.29·5-s + 7-s + 4.60·9-s − 3.92·11-s + 3.02·13-s − 3.55·15-s + 17-s − 1.45·19-s − 2.75·21-s + 2.74·23-s − 3.33·25-s − 4.43·27-s − 4.12·29-s − 5.90·31-s + 10.8·33-s + 1.29·35-s − 2.16·37-s − 8.34·39-s + 10.3·41-s − 0.144·43-s + 5.94·45-s − 0.778·47-s + 49-s − 2.75·51-s − 3.76·53-s − 5.07·55-s + ⋯ |
| L(s) = 1 | − 1.59·3-s + 0.577·5-s + 0.377·7-s + 1.53·9-s − 1.18·11-s + 0.839·13-s − 0.918·15-s + 0.242·17-s − 0.333·19-s − 0.601·21-s + 0.571·23-s − 0.667·25-s − 0.853·27-s − 0.765·29-s − 1.06·31-s + 1.88·33-s + 0.218·35-s − 0.355·37-s − 1.33·39-s + 1.62·41-s − 0.0219·43-s + 0.886·45-s − 0.113·47-s + 0.142·49-s − 0.386·51-s − 0.517·53-s − 0.683·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 - 1.29T + 5T^{2} \) |
| 11 | \( 1 + 3.92T + 11T^{2} \) |
| 13 | \( 1 - 3.02T + 13T^{2} \) |
| 19 | \( 1 + 1.45T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 4.12T + 29T^{2} \) |
| 31 | \( 1 + 5.90T + 31T^{2} \) |
| 37 | \( 1 + 2.16T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 0.144T + 43T^{2} \) |
| 47 | \( 1 + 0.778T + 47T^{2} \) |
| 53 | \( 1 + 3.76T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 2.81T + 61T^{2} \) |
| 67 | \( 1 - 2.35T + 67T^{2} \) |
| 71 | \( 1 - 9.21T + 71T^{2} \) |
| 73 | \( 1 + 1.02T + 73T^{2} \) |
| 79 | \( 1 - 6.93T + 79T^{2} \) |
| 83 | \( 1 - 2.72T + 83T^{2} \) |
| 89 | \( 1 + 3.41T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.927540017926246538252297550646, −7.34198313724730952762279945148, −6.36052424271711993221239748728, −5.82294957871924565608292091499, −5.30659272695626744166193740540, −4.62647533895079938473297889911, −3.58406429786223865831553232854, −2.26090652555490600256794219674, −1.24751590195106360674071792598, 0,
1.24751590195106360674071792598, 2.26090652555490600256794219674, 3.58406429786223865831553232854, 4.62647533895079938473297889911, 5.30659272695626744166193740540, 5.82294957871924565608292091499, 6.36052424271711993221239748728, 7.34198313724730952762279945148, 7.927540017926246538252297550646
