| L(s) = 1 | + 1.22·2-s − 1.36·3-s − 0.510·4-s − 2.60·5-s − 1.66·6-s − 5.05·7-s − 3.06·8-s − 1.13·9-s − 3.18·10-s + 1.31·11-s + 0.696·12-s − 2.43·13-s − 6.17·14-s + 3.56·15-s − 2.71·16-s − 0.535·17-s − 1.38·18-s − 5.23·19-s + 1.33·20-s + 6.90·21-s + 1.60·22-s + 4.15·23-s + 4.18·24-s + 1.79·25-s − 2.97·26-s + 5.64·27-s + 2.58·28-s + ⋯ |
| L(s) = 1 | + 0.863·2-s − 0.788·3-s − 0.255·4-s − 1.16·5-s − 0.680·6-s − 1.91·7-s − 1.08·8-s − 0.378·9-s − 1.00·10-s + 0.395·11-s + 0.201·12-s − 0.676·13-s − 1.65·14-s + 0.919·15-s − 0.679·16-s − 0.129·17-s − 0.326·18-s − 1.20·19-s + 0.297·20-s + 1.50·21-s + 0.341·22-s + 0.865·23-s + 0.854·24-s + 0.359·25-s − 0.583·26-s + 1.08·27-s + 0.487·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1527730487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1527730487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 + 2.60T + 5T^{2} \) |
| 7 | \( 1 + 5.05T + 7T^{2} \) |
| 11 | \( 1 - 1.31T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 + 0.535T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 29 | \( 1 + 5.78T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 - 0.598T + 41T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 + 6.59T + 53T^{2} \) |
| 59 | \( 1 + 9.36T + 59T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 5.13T + 71T^{2} \) |
| 73 | \( 1 - 7.47T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 6.14T + 83T^{2} \) |
| 89 | \( 1 + 6.17T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−9.201610271285179561527599423019, −8.619903999601916209145454410966, −7.39683632846014083005070324341, −6.52127638568059862737771794487, −6.10569393175128035294526640436, −5.11409977304303249901354789032, −4.26616028841880908665782575654, −3.50871462404025906942864207034, −2.82781578883842383830483578348, −0.22422670711372285098967437004,
0.22422670711372285098967437004, 2.82781578883842383830483578348, 3.50871462404025906942864207034, 4.26616028841880908665782575654, 5.11409977304303249901354789032, 6.10569393175128035294526640436, 6.52127638568059862737771794487, 7.39683632846014083005070324341, 8.619903999601916209145454410966, 9.201610271285179561527599423019
