L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 5·11-s − 12-s + 3·13-s + 14-s + 16-s + 3·17-s − 18-s + 21-s + 5·22-s − 23-s + 24-s − 3·26-s − 27-s − 28-s − 3·29-s − 10·31-s − 32-s + 5·33-s − 3·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s − 0.208·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s − 0.188·28-s − 0.557·29-s − 1.79·31-s − 0.176·32-s + 0.870·33-s − 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.69761710069155, −12.23816317602019, −11.67315416333099, −11.24386374837711, −10.83537703120756, −10.42941633227307, −10.12224841784203, −9.562655784142560, −9.152790918819532, −8.619752058516981, −8.098715926103398, −7.576460882403320, −7.445737012487133, −6.711694833583122, −6.239626637062214, −5.728976491191425, −5.413277668351892, −4.931415239997504, −4.173841484839473, −3.506313782283619, −3.265755262311128, −2.344345005942894, −2.068780741134423, −1.213861778031358, −0.6341411205832640, 0,
0.6341411205832640, 1.213861778031358, 2.068780741134423, 2.344345005942894, 3.265755262311128, 3.506313782283619, 4.173841484839473, 4.931415239997504, 5.413277668351892, 5.728976491191425, 6.239626637062214, 6.711694833583122, 7.445737012487133, 7.576460882403320, 8.098715926103398, 8.619752058516981, 9.152790918819532, 9.562655784142560, 10.12224841784203, 10.42941633227307, 10.83537703120756, 11.24386374837711, 11.67315416333099, 12.23816317602019, 12.69761710069155