| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s + 3·7-s − 8-s + 9-s − 3·10-s + 6·11-s − 12-s + 2·13-s − 3·14-s − 3·15-s + 16-s − 18-s + 3·20-s − 3·21-s − 6·22-s + 4·23-s + 24-s + 4·25-s − 2·26-s − 27-s + 3·28-s − 6·29-s + 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.80·11-s − 0.288·12-s + 0.554·13-s − 0.801·14-s − 0.774·15-s + 1/4·16-s − 0.235·18-s + 0.670·20-s − 0.654·21-s − 1.27·22-s + 0.834·23-s + 0.204·24-s + 4/5·25-s − 0.392·26-s − 0.192·27-s + 0.566·28-s − 1.11·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.573695491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.573695491\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.790113993099719615005449926489, −9.244902986697387973451868921320, −8.570993744520712513315236264075, −7.37053098708112558878260165463, −6.54768781150230855499961253121, −5.83097538657703111650312212131, −4.97298917545574338513348241661, −3.67797642946134964024333075145, −1.86881404003208925845535257123, −1.34374083667403483087824275167,
1.34374083667403483087824275167, 1.86881404003208925845535257123, 3.67797642946134964024333075145, 4.97298917545574338513348241661, 5.83097538657703111650312212131, 6.54768781150230855499961253121, 7.37053098708112558878260165463, 8.570993744520712513315236264075, 9.244902986697387973451868921320, 9.790113993099719615005449926489
