| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 11-s − 12-s + 6·13-s + 14-s + 16-s + 2·17-s − 18-s + 21-s − 22-s + 6·23-s + 24-s − 6·26-s − 27-s − 28-s − 8·31-s − 32-s − 33-s − 2·34-s + 36-s − 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.218·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s − 1.17·26-s − 0.192·27-s − 0.188·28-s − 1.43·31-s − 0.176·32-s − 0.174·33-s − 0.342·34-s + 1/6·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−16.85083043702544, −16.21682633245265, −15.71592341897202, −15.29699999999900, −14.51229447317722, −13.82016778766516, −13.21038092315594, −12.62429748944709, −12.06184002641893, −11.33031564090744, −10.91104361626004, −10.45662978629626, −9.727661926033551, −9.000976680454169, −8.696920543086640, −7.884165731138742, −7.018580333511469, −6.756924786371326, −5.825173235697487, −5.521230214985720, −4.458355224905535, −3.559878150974613, −3.078310330160699, −1.713329606816732, −1.162565552915055, 0,
1.162565552915055, 1.713329606816732, 3.078310330160699, 3.559878150974613, 4.458355224905535, 5.521230214985720, 5.825173235697487, 6.756924786371326, 7.018580333511469, 7.884165731138742, 8.696920543086640, 9.000976680454169, 9.727661926033551, 10.45662978629626, 10.91104361626004, 11.33031564090744, 12.06184002641893, 12.62429748944709, 13.21038092315594, 13.82016778766516, 14.51229447317722, 15.29699999999900, 15.71592341897202, 16.21682633245265, 16.85083043702544
