| L(s) = 1 | − 2.44·2-s + 3.95·4-s − 3.44·7-s − 4.77·8-s + 3.26·11-s − 3.23·13-s + 8.39·14-s + 3.73·16-s + 5.05·17-s − 3.08·19-s − 7.97·22-s − 1.54·23-s + 7.88·26-s − 13.6·28-s + 3.12·29-s + 7.44·31-s + 0.434·32-s − 12.3·34-s − 5.75·37-s + 7.51·38-s − 5.41·41-s + 2.53·43-s + 12.9·44-s + 3.77·46-s − 7.07·47-s + 4.83·49-s − 12.7·52-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.97·4-s − 1.30·7-s − 1.68·8-s + 0.984·11-s − 0.896·13-s + 2.24·14-s + 0.932·16-s + 1.22·17-s − 0.706·19-s − 1.69·22-s − 0.322·23-s + 1.54·26-s − 2.57·28-s + 0.579·29-s + 1.33·31-s + 0.0768·32-s − 2.11·34-s − 0.946·37-s + 1.21·38-s − 0.846·41-s + 0.385·43-s + 1.94·44-s + 0.556·46-s − 1.03·47-s + 0.690·49-s − 1.77·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5398348690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5398348690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 - 2.53T + 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 - 7.83T + 61T^{2} \) |
| 67 | \( 1 + 1.84T + 67T^{2} \) |
| 71 | \( 1 - 0.713T + 71T^{2} \) |
| 73 | \( 1 + 1.88T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 3.95T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.357427286726118049785729750540, −7.50549852309611767311555985066, −6.79996119860030214357085329812, −6.48224250106040503476141330724, −5.58363088464368628484997821597, −4.35957175708812249933243622913, −3.33090581184449465476306446440, −2.60041485805015664536286515145, −1.54477262692231714187077074971, −0.51909783592624251110818563400,
0.51909783592624251110818563400, 1.54477262692231714187077074971, 2.60041485805015664536286515145, 3.33090581184449465476306446440, 4.35957175708812249933243622913, 5.58363088464368628484997821597, 6.48224250106040503476141330724, 6.79996119860030214357085329812, 7.50549852309611767311555985066, 8.357427286726118049785729750540
