| L(s) = 1 | − 5-s − 4·11-s − 13-s − 6·17-s + 8·23-s + 25-s − 6·29-s + 4·31-s − 2·37-s − 10·41-s − 4·43-s − 8·47-s + 2·53-s + 4·55-s + 12·59-s + 2·61-s + 65-s − 16·67-s + 8·71-s + 6·73-s − 16·79-s + 4·83-s + 6·85-s − 2·89-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.277·13-s − 1.45·17-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 0.274·53-s + 0.539·55-s + 1.56·59-s + 0.256·61-s + 0.124·65-s − 1.95·67-s + 0.949·71-s + 0.702·73-s − 1.80·79-s + 0.439·83-s + 0.650·85-s − 0.211·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.44009678283794, −12.95583836737886, −12.65636509821119, −11.80262551790354, −11.60989335126266, −11.06811232833462, −10.65775652530524, −10.21365454470135, −9.717543551249568, −9.074329493595341, −8.683434614363055, −8.279301690802333, −7.700627651125801, −7.271894085141727, −6.598126405810570, −6.576757909573541, −5.480568510359264, −5.141983114884470, −4.831620289279219, −4.110732224854451, −3.610363629421629, −2.858871200453694, −2.593853163042306, −1.829541190186967, −1.151484402740541, 0, 0,
1.151484402740541, 1.829541190186967, 2.593853163042306, 2.858871200453694, 3.610363629421629, 4.110732224854451, 4.831620289279219, 5.141983114884470, 5.480568510359264, 6.576757909573541, 6.598126405810570, 7.271894085141727, 7.700627651125801, 8.279301690802333, 8.683434614363055, 9.074329493595341, 9.717543551249568, 10.21365454470135, 10.65775652530524, 11.06811232833462, 11.60989335126266, 11.80262551790354, 12.65636509821119, 12.95583836737886, 13.44009678283794
