| L(s) = 1 | + 5·5-s − 35.5·7-s + 44.7·11-s + 77.8·13-s − 120.·17-s − 121.·19-s − 152.·23-s + 25·25-s + 187.·29-s + 161.·31-s − 177.·35-s + 13.3·37-s + 188.·41-s + 81.5·43-s − 48.6·47-s + 923.·49-s − 707.·53-s + 223.·55-s − 16.4·59-s − 743.·61-s + 389.·65-s + 59.2·67-s − 144.·71-s + 657.·73-s − 1.59e3·77-s + 454.·79-s + 165.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.92·7-s + 1.22·11-s + 1.66·13-s − 1.71·17-s − 1.46·19-s − 1.38·23-s + 0.200·25-s + 1.20·29-s + 0.934·31-s − 0.859·35-s + 0.0592·37-s + 0.716·41-s + 0.289·43-s − 0.150·47-s + 2.69·49-s − 1.83·53-s + 0.548·55-s − 0.0363·59-s − 1.56·61-s + 0.742·65-s + 0.108·67-s − 0.241·71-s + 1.05·73-s − 2.35·77-s + 0.646·79-s + 0.218·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.594192109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.594192109\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5T \) |
| good | 7 | \( 1 + 35.5T + 343T^{2} \) |
| 11 | \( 1 - 44.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 77.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 13.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 188.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 81.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 48.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 707.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 16.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 743.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 59.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 144.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 657.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 454.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 165.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 535.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 436.T + 9.12e5T^{2} \) |
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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
−8.995443231769473771126914036239, −8.141806120930525218885359744977, −6.62464073526742028746026769065, −6.32110421439244311670709599002, −6.17470739182867370858944161155, −4.39497490970996373980727478713, −3.87416039637715735620895897879, −2.90053962886287912153544307598, −1.85017929530678248274301471379, −0.56023213289141030148195971025,
0.56023213289141030148195971025, 1.85017929530678248274301471379, 2.90053962886287912153544307598, 3.87416039637715735620895897879, 4.39497490970996373980727478713, 6.17470739182867370858944161155, 6.32110421439244311670709599002, 6.62464073526742028746026769065, 8.141806120930525218885359744977, 8.995443231769473771126914036239
