L(s) = 1 | + 2-s − 3-s + 4-s − 4·5-s − 6-s + 3·7-s + 8-s − 2·9-s − 4·10-s + 2·11-s − 12-s − 13-s + 3·14-s + 4·15-s + 16-s + 3·17-s − 2·18-s − 19-s − 4·20-s − 3·21-s + 2·22-s − 23-s − 24-s + 11·25-s − 26-s + 5·27-s + 3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s − 1.26·10-s + 0.603·11-s − 0.288·12-s − 0.277·13-s + 0.801·14-s + 1.03·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 0.229·19-s − 0.894·20-s − 0.654·21-s + 0.426·22-s − 0.208·23-s − 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.962·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8192456330\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8192456330\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−16.27742004806018738020850312917, −15.00174295764739089605520124901, −14.37348632970750952303967845812, −12.38825742592724708014161209088, −11.61515083240032291487198991803, −10.98850264170077345648603277341, −8.408486909011940238981299566497, −7.25038064772684706053554512263, −5.28069074518443602737935578596, −3.86057015395193111701613139713,
3.86057015395193111701613139713, 5.28069074518443602737935578596, 7.25038064772684706053554512263, 8.408486909011940238981299566497, 10.98850264170077345648603277341, 11.61515083240032291487198991803, 12.38825742592724708014161209088, 14.37348632970750952303967845812, 15.00174295764739089605520124901, 16.27742004806018738020850312917