| 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 + 4·17-s − 18-s − 2·19-s − 21-s + 22-s + 8·23-s − 24-s − 6·26-s + 27-s − 28-s − 6·29-s + 6·31-s − 32-s − 33-s − 4·34-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 + 0.970·17-s − 0.235·18-s − 0.458·19-s − 0.218·21-s + 0.213·22-s + 1.66·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + 1.07·31-s − 0.176·32-s − 0.174·33-s − 0.685·34-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
| \(L(1)\) |
\(\approx\) |
\(2.200735447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200735447\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.38059917860669, −15.95818127869885, −15.38257544987002, −14.77731331701063, −14.32586639644554, −13.42091828789815, −13.01682721569070, −12.65575312270812, −11.49058313839879, −11.29821116237929, −10.48042268841516, −10.02837212585424, −9.254354481536487, −8.868930190457864, −8.236350916128656, −7.668063734637433, −7.047162879344115, −6.224122176484179, −5.811742508445882, −4.788480719355793, −3.869846045345714, −3.212050065584354, −2.599526594185449, −1.490113523608533, −0.8000528519545730,
0.8000528519545730, 1.490113523608533, 2.599526594185449, 3.212050065584354, 3.869846045345714, 4.788480719355793, 5.811742508445882, 6.224122176484179, 7.047162879344115, 7.668063734637433, 8.236350916128656, 8.868930190457864, 9.254354481536487, 10.02837212585424, 10.48042268841516, 11.29821116237929, 11.49058313839879, 12.65575312270812, 13.01682721569070, 13.42091828789815, 14.32586639644554, 14.77731331701063, 15.38257544987002, 15.95818127869885, 16.38059917860669
