| L(s) = 1 | − 2-s + 2·3-s + 4-s + 2·5-s − 2·6-s + 7-s − 8-s + 9-s − 2·10-s − 2·11-s + 2·12-s + 4·13-s − 14-s + 4·15-s + 16-s + 6·17-s − 18-s − 6·19-s + 2·20-s + 2·21-s + 2·22-s + 8·23-s − 2·24-s − 25-s − 4·26-s − 4·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.603·11-s + 0.577·12-s + 1.10·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.37·19-s + 0.447·20-s + 0.436·21-s + 0.426·22-s + 1.66·23-s − 0.408·24-s − 1/5·25-s − 0.784·26-s − 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.841787016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.841787016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.65481602424905054585459165545, −9.527576323384131864698728777028, −9.062357874649631789347759686870, −8.166126801585589938617752867726, −7.57077576103548182098351353000, −6.24929390141495861251889858166, −5.36289825441543847775925555908, −3.67065023610212490952651051429, −2.59941261306384637349285551759, −1.53868060877101345031490281977,
1.53868060877101345031490281977, 2.59941261306384637349285551759, 3.67065023610212490952651051429, 5.36289825441543847775925555908, 6.24929390141495861251889858166, 7.57077576103548182098351353000, 8.166126801585589938617752867726, 9.062357874649631789347759686870, 9.527576323384131864698728777028, 10.65481602424905054585459165545
