| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 12-s − 2·13-s − 15-s + 16-s + 4·17-s + 18-s − 8·19-s − 20-s + 23-s + 24-s + 25-s − 2·26-s + 27-s − 30-s + 4·31-s + 32-s + 4·34-s + 36-s + 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.182·30-s + 0.718·31-s + 0.176·32-s + 0.685·34-s + 1/6·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.89326833905503, −14.70735752550728, −14.50298538882800, −13.60459049292722, −13.14258829109679, −12.71114511449689, −12.21514078571117, −11.68036591304992, −11.06101874456672, −10.55252617066938, −9.872677506989631, −9.506727015043750, −8.553093508905625, −8.201561293279492, −7.707800689105774, −6.972837793585321, −6.530971147457800, −5.849714889078727, −5.132479155671697, −4.393165293985580, −4.150812302757528, −3.227265639776776, −2.794376448869587, −2.036607585552266, −1.215866057774804, 0,
1.215866057774804, 2.036607585552266, 2.794376448869587, 3.227265639776776, 4.150812302757528, 4.393165293985580, 5.132479155671697, 5.849714889078727, 6.530971147457800, 6.972837793585321, 7.707800689105774, 8.201561293279492, 8.553093508905625, 9.506727015043750, 9.872677506989631, 10.55252617066938, 11.06101874456672, 11.68036591304992, 12.21514078571117, 12.71114511449689, 13.14258829109679, 13.60459049292722, 14.50298538882800, 14.70735752550728, 14.89326833905503
