| L(s) = 1 | − 5·5-s − 16.4·7-s + 57.6·11-s + 38.7·13-s − 31.4·17-s − 70.6·19-s − 7.37·23-s + 25·25-s + 17.1·29-s + 50.9·31-s + 82.0·35-s − 159.·37-s + 63.2·41-s + 84.6·43-s + 434.·47-s − 73.6·49-s − 138.·53-s − 288.·55-s − 631.·59-s + 82.8·61-s − 193.·65-s + 431.·67-s + 450.·71-s − 33.5·73-s − 945.·77-s − 509.·79-s − 811.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.886·7-s + 1.57·11-s + 0.827·13-s − 0.448·17-s − 0.852·19-s − 0.0668·23-s + 0.200·25-s + 0.109·29-s + 0.295·31-s + 0.396·35-s − 0.709·37-s + 0.240·41-s + 0.300·43-s + 1.34·47-s − 0.214·49-s − 0.358·53-s − 0.706·55-s − 1.39·59-s + 0.173·61-s − 0.370·65-s + 0.787·67-s + 0.752·71-s − 0.0538·73-s − 1.39·77-s − 0.725·79-s − 1.07·83-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
| \(L(2)\) |
\(\approx\) |
\(1.732927230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732927230\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 16.4T + 343T^{2} \) |
| 11 | \( 1 - 57.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.37T + 1.21e4T^{2} \) |
| 29 | \( 1 - 17.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 50.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 159.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 63.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 84.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 434.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 138.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 631.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 82.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 431.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 450.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 33.5T + 3.89e5T^{2} \) |
| 79 | \( 1 + 509.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 811.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 499.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 625.T + 9.12e5T^{2} \) |
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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
−8.870910126099076503845197778110, −8.043380851633056680555442126914, −6.91874678248098538665194116291, −6.50762781886380997598705549825, −5.77052802858411494790863932836, −4.40558758761880139655108536113, −3.87530596506929639928862302611, −3.02665822920982563008422407658, −1.72474942224237824769599321239, −0.60588601962814371971005132149,
0.60588601962814371971005132149, 1.72474942224237824769599321239, 3.02665822920982563008422407658, 3.87530596506929639928862302611, 4.40558758761880139655108536113, 5.77052802858411494790863932836, 6.50762781886380997598705549825, 6.91874678248098538665194116291, 8.043380851633056680555442126914, 8.870910126099076503845197778110
