| L(s) = 1 | + 5-s − 7-s − 3·9-s − 6·11-s − 6·13-s + 7·17-s − 2·19-s − 23-s + 25-s + 5·29-s + 31-s − 35-s + 5·37-s − 7·41-s − 8·43-s − 3·45-s + 8·47-s − 6·49-s − 3·53-s − 6·55-s − 13·59-s + 8·61-s + 3·63-s − 6·65-s + 9·67-s + 7·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s − 1.80·11-s − 1.66·13-s + 1.69·17-s − 0.458·19-s − 0.208·23-s + 1/5·25-s + 0.928·29-s + 0.179·31-s − 0.169·35-s + 0.821·37-s − 1.09·41-s − 1.21·43-s − 0.447·45-s + 1.16·47-s − 6/7·49-s − 0.412·53-s − 0.809·55-s − 1.69·59-s + 1.02·61-s + 0.377·63-s − 0.744·65-s + 1.09·67-s + 0.830·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9796751310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9796751310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.068925435810957361782031297715, −7.28140079285334238457724310428, −6.48179794069436181095842315364, −5.57899546883210075167832836854, −5.28635578852051639058136561201, −4.55929981386272476853826731364, −3.12995174290444779672520347917, −2.85283570837514386406663441317, −2.01160089877547536255094536472, −0.46418001080947767782019002876,
0.46418001080947767782019002876, 2.01160089877547536255094536472, 2.85283570837514386406663441317, 3.12995174290444779672520347917, 4.55929981386272476853826731364, 5.28635578852051639058136561201, 5.57899546883210075167832836854, 6.48179794069436181095842315364, 7.28140079285334238457724310428, 8.068925435810957361782031297715
