| L(s) = 1 | + 2·5-s − 11-s − 2·13-s + 6·17-s − 4·23-s − 25-s − 2·29-s − 10·37-s + 6·41-s − 8·43-s − 4·47-s + 6·53-s − 2·55-s − 12·59-s − 2·61-s − 4·65-s + 4·67-s − 12·71-s + 14·73-s + 16·79-s − 12·83-s + 12·85-s + 10·89-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.301·11-s − 0.554·13-s + 1.45·17-s − 0.834·23-s − 1/5·25-s − 0.371·29-s − 1.64·37-s + 0.937·41-s − 1.21·43-s − 0.583·47-s + 0.824·53-s − 0.269·55-s − 1.56·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s − 1.42·71-s + 1.63·73-s + 1.80·79-s − 1.31·83-s + 1.30·85-s + 1.05·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.229303826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.229303826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.76645702868037, −14.22610443492003, −13.79446159543252, −13.40392319374763, −12.69671735597790, −12.17466558191656, −11.89912532913077, −11.06891217509349, −10.44008832021382, −10.05903917141630, −9.618052891222573, −9.115008933564385, −8.349388114192567, −7.806677788455435, −7.359588186222068, −6.580954408200086, −6.052821793584637, −5.402400829805340, −5.135435156327870, −4.248649058725036, −3.477719500980244, −2.938538761854423, −2.014413372611371, −1.643332281869820, −0.5298136100197080,
0.5298136100197080, 1.643332281869820, 2.014413372611371, 2.938538761854423, 3.477719500980244, 4.248649058725036, 5.135435156327870, 5.402400829805340, 6.052821793584637, 6.580954408200086, 7.359588186222068, 7.806677788455435, 8.349388114192567, 9.115008933564385, 9.618052891222573, 10.05903917141630, 10.44008832021382, 11.06891217509349, 11.89912532913077, 12.17466558191656, 12.69671735597790, 13.40392319374763, 13.79446159543252, 14.22610443492003, 14.76645702868037
