| L(s) = 1 | − 5·5-s − 11.6·7-s − 47.4·11-s + 84.2·13-s + 26.9·17-s + 152.·19-s − 177.·23-s + 25·25-s − 60.0·29-s + 92.1·31-s + 58.1·35-s − 221.·37-s − 115.·41-s − 383.·43-s + 317.·47-s − 207.·49-s + 257.·53-s + 237.·55-s − 642.·59-s − 662.·61-s − 421.·65-s − 597.·67-s − 500.·71-s + 989.·73-s + 552.·77-s + 517.·79-s − 605.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.627·7-s − 1.30·11-s + 1.79·13-s + 0.384·17-s + 1.83·19-s − 1.60·23-s + 0.200·25-s − 0.384·29-s + 0.533·31-s + 0.280·35-s − 0.985·37-s − 0.438·41-s − 1.36·43-s + 0.984·47-s − 0.605·49-s + 0.667·53-s + 0.582·55-s − 1.41·59-s − 1.38·61-s − 0.804·65-s − 1.08·67-s − 0.836·71-s + 1.58·73-s + 0.817·77-s + 0.737·79-s − 0.801·83-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
| \(L(2)\) |
\(\approx\) |
\(1.480511205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.480511205\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 11.6T + 343T^{2} \) |
| 11 | \( 1 + 47.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 60.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 92.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 115.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 317.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 257.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 642.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 662.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 597.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 500.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 989.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 517.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 605.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 742.T + 9.12e5T^{2} \) |
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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
−8.603195329745304412067047236171, −7.930585679341966779541506241138, −7.33926244443408543604324137248, −6.22872659077397790481094885941, −5.67890031403640141161490368175, −4.73111774103424182475517690557, −3.49416434873689011936539627471, −3.20721752477552417217794960705, −1.74674648865907279351814150715, −0.55139443671118230987393666572,
0.55139443671118230987393666572, 1.74674648865907279351814150715, 3.20721752477552417217794960705, 3.49416434873689011936539627471, 4.73111774103424182475517690557, 5.67890031403640141161490368175, 6.22872659077397790481094885941, 7.33926244443408543604324137248, 7.930585679341966779541506241138, 8.603195329745304412067047236171
