| L(s) = 1 | + 5-s − 4·7-s − 3·9-s − 6·13-s + 6·17-s − 4·19-s + 4·23-s + 25-s + 2·29-s + 8·31-s − 4·35-s − 10·37-s − 10·41-s − 3·45-s + 4·47-s + 9·49-s − 10·53-s − 4·59-s + 2·61-s + 12·63-s − 6·65-s − 8·67-s + 14·73-s + 16·79-s + 9·81-s + 8·83-s + 6·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 9-s − 1.66·13-s + 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.676·35-s − 1.64·37-s − 1.56·41-s − 0.447·45-s + 0.583·47-s + 9/7·49-s − 1.37·53-s − 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.744·65-s − 0.977·67-s + 1.63·73-s + 1.80·79-s + 81-s + 0.878·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.024952531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.024952531\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.340840132847719622062692308486, −7.48616190366262769632803117816, −6.66746387831112665940083046403, −6.23847770530914576795854369495, −5.32966433924218422966082211753, −4.78881472693185418178908094230, −3.37097589433260462955017179248, −3.02287211325894299162157018527, −2.11428900783813517594963447864, −0.52309490573401401510532722199,
0.52309490573401401510532722199, 2.11428900783813517594963447864, 3.02287211325894299162157018527, 3.37097589433260462955017179248, 4.78881472693185418178908094230, 5.32966433924218422966082211753, 6.23847770530914576795854369495, 6.66746387831112665940083046403, 7.48616190366262769632803117816, 8.340840132847719622062692308486
