L(s) = 1 | + 5-s + 2·7-s + 11-s − 13-s + 6·17-s − 8·19-s − 8·23-s + 25-s − 4·29-s + 2·31-s + 2·35-s − 8·37-s + 6·41-s − 12·43-s + 8·47-s − 3·49-s − 12·53-s + 55-s + 8·59-s + 6·61-s − 65-s + 12·67-s + 8·71-s + 2·73-s + 2·77-s + 8·79-s + 12·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.301·11-s − 0.277·13-s + 1.45·17-s − 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.742·29-s + 0.359·31-s + 0.338·35-s − 1.31·37-s + 0.937·41-s − 1.82·43-s + 1.16·47-s − 3/7·49-s − 1.64·53-s + 0.134·55-s + 1.04·59-s + 0.768·61-s − 0.124·65-s + 1.46·67-s + 0.949·71-s + 0.234·73-s + 0.227·77-s + 0.900·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.319820625\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.319820625\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.88681658214657, −13.23735427306222, −12.66137662124560, −12.24918298508170, −11.91286161965297, −11.18373928545977, −10.83403627535119, −10.21354169421551, −9.839612501196271, −9.401127637534945, −8.625736363969273, −8.172577657273952, −7.943099873490665, −7.197305524065002, −6.527116327834943, −6.212569909998951, −5.444946808850647, −5.154244128929725, −4.427709848781718, −3.830354541063374, −3.402190956959455, −2.350988297211355, −1.998539313101561, −1.418891538259011, −0.4636084179166856,
0.4636084179166856, 1.418891538259011, 1.998539313101561, 2.350988297211355, 3.402190956959455, 3.830354541063374, 4.427709848781718, 5.154244128929725, 5.444946808850647, 6.212569909998951, 6.527116327834943, 7.197305524065002, 7.943099873490665, 8.172577657273952, 8.625736363969273, 9.401127637534945, 9.839612501196271, 10.21354169421551, 10.83403627535119, 11.18373928545977, 11.91286161965297, 12.24918298508170, 12.66137662124560, 13.23735427306222, 13.88681658214657