L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 7-s − 3·8-s + 9-s − 10-s − 12-s − 6·13-s − 14-s − 15-s − 16-s − 2·17-s + 18-s + 8·19-s + 20-s − 21-s − 3·24-s + 25-s − 6·26-s + 27-s + 28-s − 2·29-s − 30-s + 4·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.218·21-s − 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.58003990179407, −14.14643208291135, −13.64513682053911, −13.22712775412256, −12.71811308425447, −12.14032151208372, −11.82798048467272, −11.36888783394944, −10.29358187451852, −10.02642414126790, −9.348370047330654, −9.161511482376572, −8.390011093702788, −7.840342444117310, −7.270990171136421, −6.892431849307683, −6.030303025856026, −5.454600388976996, −4.767198814607318, −4.552817773607729, −3.733653270756340, −3.123646354162737, −2.824149464454914, −1.949071054053506, −0.8371493983340824, 0,
0.8371493983340824, 1.949071054053506, 2.824149464454914, 3.123646354162737, 3.733653270756340, 4.552817773607729, 4.767198814607318, 5.454600388976996, 6.030303025856026, 6.892431849307683, 7.270990171136421, 7.840342444117310, 8.390011093702788, 9.161511482376572, 9.348370047330654, 10.02642414126790, 10.29358187451852, 11.36888783394944, 11.82798048467272, 12.14032151208372, 12.71811308425447, 13.22712775412256, 13.64513682053911, 14.14643208291135, 14.58003990179407