| L(s) = 1 | − 2-s + 4-s − 2.04·5-s + 2.45·7-s − 8-s + 2.04·10-s − 0.203·11-s + 3.95·13-s − 2.45·14-s + 16-s − 2.98·17-s + 6.08·19-s − 2.04·20-s + 0.203·22-s + 0.149·23-s − 0.837·25-s − 3.95·26-s + 2.45·28-s − 1.18·29-s + 1.40·31-s − 32-s + 2.98·34-s − 5.00·35-s − 3.56·37-s − 6.08·38-s + 2.04·40-s + 4.57·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.912·5-s + 0.926·7-s − 0.353·8-s + 0.645·10-s − 0.0613·11-s + 1.09·13-s − 0.655·14-s + 0.250·16-s − 0.724·17-s + 1.39·19-s − 0.456·20-s + 0.0433·22-s + 0.0312·23-s − 0.167·25-s − 0.775·26-s + 0.463·28-s − 0.219·29-s + 0.252·31-s − 0.176·32-s + 0.512·34-s − 0.845·35-s − 0.586·37-s − 0.987·38-s + 0.322·40-s + 0.714·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.405263827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.405263827\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 2.04T + 5T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 + 0.203T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 19 | \( 1 - 6.08T + 19T^{2} \) |
| 23 | \( 1 - 0.149T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 - 1.40T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 - 4.57T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 + 8.68T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 8.79T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 - 2.21T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 8.22T + 97T^{2} \) |
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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
−7.81474196318734487050672329767, −7.47498729575921053628583176393, −6.60906243074733462983494156254, −5.81777043366691043272458544693, −5.04782517660248997652721459945, −4.16736054987755045209836117641, −3.55562205099824796778178727838, −2.56306122269191700701819619723, −1.54003626322720291778468439485, −0.69515067842685968119980432254,
0.69515067842685968119980432254, 1.54003626322720291778468439485, 2.56306122269191700701819619723, 3.55562205099824796778178727838, 4.16736054987755045209836117641, 5.04782517660248997652721459945, 5.81777043366691043272458544693, 6.60906243074733462983494156254, 7.47498729575921053628583176393, 7.81474196318734487050672329767
