L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 4·5-s + 2·6-s + 7-s − 2·9-s + 8·10-s + 2·11-s − 2·12-s + 4·13-s − 2·14-s + 4·15-s − 4·16-s − 2·17-s + 4·18-s + 5·19-s − 8·20-s − 21-s − 4·22-s + 9·23-s + 11·25-s − 8·26-s + 5·27-s + 2·28-s − 29-s − 8·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 1.78·5-s + 0.816·6-s + 0.377·7-s − 2/3·9-s + 2.52·10-s + 0.603·11-s − 0.577·12-s + 1.10·13-s − 0.534·14-s + 1.03·15-s − 16-s − 0.485·17-s + 0.942·18-s + 1.14·19-s − 1.78·20-s − 0.218·21-s − 0.852·22-s + 1.87·23-s + 11/5·25-s − 1.56·26-s + 0.962·27-s + 0.377·28-s − 0.185·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3505417174\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3505417174\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−11.71095812379068084447954858997, −11.33602129559791134643564976933, −10.79131551663212647721316181454, −9.142061565933134180456079240781, −8.518787779003785407287218429384, −7.61819623880598528187301543377, −6.71652888402919591077959675143, −4.95184979272712106895823280322, −3.51539385012532972520182576397, −0.858748240911086982856794370077,
0.858748240911086982856794370077, 3.51539385012532972520182576397, 4.95184979272712106895823280322, 6.71652888402919591077959675143, 7.61819623880598528187301543377, 8.518787779003785407287218429384, 9.142061565933134180456079240781, 10.79131551663212647721316181454, 11.33602129559791134643564976933, 11.71095812379068084447954858997