| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s + 13-s + 15-s + 16-s + 2·17-s − 18-s − 19-s − 20-s − 8·23-s + 24-s + 25-s − 26-s − 27-s + 6·29-s − 30-s − 32-s − 2·34-s + 36-s − 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s − 0.176·32-s − 0.342·34-s + 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.55132083659855, −12.01746396854015, −11.91515880446654, −11.43502473990821, −10.83770574934808, −10.44014333174137, −10.07339949989523, −9.766157298148237, −8.980210938497104, −8.663287142314145, −8.163359113991154, −7.760375033765256, −7.272933689736361, −6.757279312968909, −6.334266016500687, −5.833542292641413, −5.378629778901544, −4.804650531886738, −4.058669437882931, −3.888488829583171, −3.078098703627998, −2.554473796403325, −1.832902127452254, −1.317909680264730, −0.6192522636055691, 0,
0.6192522636055691, 1.317909680264730, 1.832902127452254, 2.554473796403325, 3.078098703627998, 3.888488829583171, 4.058669437882931, 4.804650531886738, 5.378629778901544, 5.833542292641413, 6.334266016500687, 6.757279312968909, 7.272933689736361, 7.760375033765256, 8.163359113991154, 8.663287142314145, 8.980210938497104, 9.766157298148237, 10.07339949989523, 10.44014333174137, 10.83770574934808, 11.43502473990821, 11.91515880446654, 12.01746396854015, 12.55132083659855
