| L(s) = 1 | + 3·3-s + 6.73·5-s + 9·9-s − 53.6·11-s − 1.73·13-s + 20.2·15-s + 46.9·17-s + 20.1·19-s + 118.·23-s − 79.6·25-s + 27·27-s − 103.·29-s + 157.·31-s − 161.·33-s − 37.7·37-s − 5.20·39-s + 287.·41-s + 504.·43-s + 60.6·45-s + 220.·47-s + 140.·51-s + 292.·53-s − 361.·55-s + 60.4·57-s + 595.·59-s + 265.·61-s − 11.6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.602·5-s + 0.333·9-s − 1.47·11-s − 0.0370·13-s + 0.347·15-s + 0.669·17-s + 0.243·19-s + 1.07·23-s − 0.636·25-s + 0.192·27-s − 0.663·29-s + 0.912·31-s − 0.849·33-s − 0.167·37-s − 0.0213·39-s + 1.09·41-s + 1.79·43-s + 0.200·45-s + 0.684·47-s + 0.386·51-s + 0.757·53-s − 0.886·55-s + 0.140·57-s + 1.31·59-s + 0.557·61-s − 0.0223·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.869417029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.869417029\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6.73T + 125T^{2} \) |
| 11 | \( 1 + 53.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 1.73T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 37.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 504.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 220.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 292.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 595.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 265.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 936.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 545.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 299.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 940.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 611.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−9.452331890825750095213847559836, −8.587365812625692662943565292757, −7.73627051430992412765991335120, −7.14694214426008729710908493761, −5.84249726426091791281948342177, −5.27995147384573481481106426645, −4.12298862875252201287046765909, −2.92315923466615363179061498661, −2.24701046416547143782900300533, −0.852267769703980784789627136244,
0.852267769703980784789627136244, 2.24701046416547143782900300533, 2.92315923466615363179061498661, 4.12298862875252201287046765909, 5.27995147384573481481106426645, 5.84249726426091791281948342177, 7.14694214426008729710908493761, 7.73627051430992412765991335120, 8.587365812625692662943565292757, 9.452331890825750095213847559836
