| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·5-s − 2·6-s + 9-s − 4·10-s + 5·11-s − 2·12-s − 13-s + 2·15-s − 4·16-s − 17-s + 2·18-s − 6·19-s − 4·20-s + 10·22-s − 4·23-s − 25-s − 2·26-s − 27-s + 8·29-s + 4·30-s + 31-s − 8·32-s − 5·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.894·5-s − 0.816·6-s + 1/3·9-s − 1.26·10-s + 1.50·11-s − 0.577·12-s − 0.277·13-s + 0.516·15-s − 16-s − 0.242·17-s + 0.471·18-s − 1.37·19-s − 0.894·20-s + 2.13·22-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.48·29-s + 0.730·30-s + 0.179·31-s − 1.41·32-s − 0.870·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.276620091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.276620091\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.79789729039036, −14.65395756142947, −13.93576496748881, −13.55499756154794, −12.78978939930530, −12.30769732772658, −11.96664504762622, −11.69039634230206, −10.99136856386567, −10.54168071438841, −9.727445973125200, −9.065329753915546, −8.521912573548510, −7.836721245978190, −7.049201562384342, −6.589551469748986, −6.112079676551136, −5.635582846337481, −4.605589963104482, −4.336704377012972, −4.018684266431482, −3.247317924433207, −2.455301975108319, −1.588588304597067, −0.4627658944599375,
0.4627658944599375, 1.588588304597067, 2.455301975108319, 3.247317924433207, 4.018684266431482, 4.336704377012972, 4.605589963104482, 5.635582846337481, 6.112079676551136, 6.589551469748986, 7.049201562384342, 7.836721245978190, 8.521912573548510, 9.065329753915546, 9.727445973125200, 10.54168071438841, 10.99136856386567, 11.69039634230206, 11.96664504762622, 12.30769732772658, 12.78978939930530, 13.55499756154794, 13.93576496748881, 14.65395756142947, 14.79789729039036
