| L(s) = 1 | + 2-s + 3-s + 4-s + 1.61·5-s + 6-s + 8-s + 9-s + 1.61·10-s − 11-s + 12-s + 1.61·15-s + 16-s + 7.34·17-s + 18-s − 1.17·19-s + 1.61·20-s − 22-s + 3.55·23-s + 24-s − 2.39·25-s + 27-s + 10.3·29-s + 1.61·30-s − 6.34·31-s + 32-s − 33-s + 7.34·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.721·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.510·10-s − 0.301·11-s + 0.288·12-s + 0.416·15-s + 0.250·16-s + 1.78·17-s + 0.235·18-s − 0.268·19-s + 0.360·20-s − 0.213·22-s + 0.741·23-s + 0.204·24-s − 0.479·25-s + 0.192·27-s + 1.93·29-s + 0.294·30-s − 1.13·31-s + 0.176·32-s − 0.174·33-s + 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.473187634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.473187634\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 1.61T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 3.55T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 + 4.94T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 6.78T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 4.39T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 + 5.05T + 83T^{2} \) |
| 89 | \( 1 - 0.773T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−8.455519488838681003497611920736, −7.990900185347315658325822616636, −7.01963751846613370843709631974, −6.40022878501910480091516913955, −5.40965087747041634572870630477, −5.00320645232063991520334346680, −3.79366666298471583255980512330, −3.12169860921723803499547760141, −2.24364888084564728255154563868, −1.22960168349114243527410817194,
1.22960168349114243527410817194, 2.24364888084564728255154563868, 3.12169860921723803499547760141, 3.79366666298471583255980512330, 5.00320645232063991520334346680, 5.40965087747041634572870630477, 6.40022878501910480091516913955, 7.01963751846613370843709631974, 7.990900185347315658325822616636, 8.455519488838681003497611920736
