| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 3·11-s − 12-s + 3·13-s − 14-s − 15-s + 16-s + 4·17-s + 18-s + 7·19-s + 20-s + 21-s + 3·22-s − 7·23-s − 24-s + 25-s + 3·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s − 0.288·12-s + 0.832·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.60·19-s + 0.223·20-s + 0.218·21-s + 0.639·22-s − 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.01639571753253, −12.34466487199586, −11.99234095771477, −11.78910599554134, −11.30992749258505, −10.65393871444521, −10.22424335795757, −9.888991303908237, −9.352911100415832, −8.906849385446339, −8.258196909048428, −7.623408691662791, −7.312967165796228, −6.584688740697176, −6.342075432348577, −5.787894314177358, −5.431520809648130, −5.036493057464930, −4.179474832613399, −3.882017331165500, −3.250249848699126, −2.953269775937924, −1.881919093877243, −1.510987473852590, −0.9819278641892859, 0,
0.9819278641892859, 1.510987473852590, 1.881919093877243, 2.953269775937924, 3.250249848699126, 3.882017331165500, 4.179474832613399, 5.036493057464930, 5.431520809648130, 5.787894314177358, 6.342075432348577, 6.584688740697176, 7.312967165796228, 7.623408691662791, 8.258196909048428, 8.906849385446339, 9.352911100415832, 9.888991303908237, 10.22424335795757, 10.65393871444521, 11.30992749258505, 11.78910599554134, 11.99234095771477, 12.34466487199586, 13.01639571753253
