| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s + 13-s + 16-s + 3·17-s − 2·19-s + 6·22-s − 26-s − 6·29-s + 4·31-s − 32-s − 3·34-s + 7·37-s + 2·38-s + 43-s − 6·44-s − 3·47-s + 52-s + 6·58-s − 6·59-s − 8·61-s − 4·62-s + 64-s − 14·67-s + 3·68-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 0.277·13-s + 1/4·16-s + 0.727·17-s − 0.458·19-s + 1.27·22-s − 0.196·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s + 1.15·37-s + 0.324·38-s + 0.152·43-s − 0.904·44-s − 0.437·47-s + 0.138·52-s + 0.787·58-s − 0.781·59-s − 1.02·61-s − 0.508·62-s + 1/8·64-s − 1.71·67-s + 0.363·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4528361624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4528361624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.86245939797886, −12.20207245103713, −11.78497345938778, −11.18260924557661, −10.77306052928358, −10.47984262588858, −10.00820439046329, −9.461365791700782, −9.163241432846129, −8.405411336728777, −8.045081173822213, −7.755169566508523, −7.309873321653619, −6.681562422327613, −6.106782368398046, −5.646939379279990, −5.275605874634425, −4.569206076632948, −4.104886814066737, −3.253094417791877, −2.874868445894739, −2.384989281253457, −1.691474395856619, −1.089633830226674, −0.2137988826278011,
0.2137988826278011, 1.089633830226674, 1.691474395856619, 2.384989281253457, 2.874868445894739, 3.253094417791877, 4.104886814066737, 4.569206076632948, 5.275605874634425, 5.646939379279990, 6.106782368398046, 6.681562422327613, 7.309873321653619, 7.755169566508523, 8.045081173822213, 8.405411336728777, 9.163241432846129, 9.461365791700782, 10.00820439046329, 10.47984262588858, 10.77306052928358, 11.18260924557661, 11.78497345938778, 12.20207245103713, 12.86245939797886
