| L(s) = 1 | − 18.7·5-s + 19.7·7-s − 5.84·11-s − 33.7·13-s − 33.2·17-s + 102.·19-s + 130.·23-s + 228.·25-s − 111.·29-s + 257.·31-s − 370.·35-s − 424.·37-s + 10.9·41-s + 324.·43-s − 599.·47-s + 45.7·49-s − 106.·53-s + 109.·55-s + 617.·59-s + 325.·61-s + 634.·65-s − 240.·67-s + 123.·71-s − 46.7·73-s − 115.·77-s − 547.·79-s − 1.21e3·83-s + ⋯ |
| L(s) = 1 | − 1.68·5-s + 1.06·7-s − 0.160·11-s − 0.719·13-s − 0.474·17-s + 1.24·19-s + 1.17·23-s + 1.82·25-s − 0.715·29-s + 1.49·31-s − 1.78·35-s − 1.88·37-s + 0.0416·41-s + 1.15·43-s − 1.86·47-s + 0.133·49-s − 0.276·53-s + 0.269·55-s + 1.36·59-s + 0.682·61-s + 1.21·65-s − 0.437·67-s + 0.207·71-s − 0.0749·73-s − 0.170·77-s − 0.779·79-s − 1.60·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 18.7T + 125T^{2} \) |
| 7 | \( 1 - 19.7T + 343T^{2} \) |
| 11 | \( 1 + 5.84T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 111.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 257.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 424.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 10.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 324.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 599.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 106.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 617.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 325.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 240.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 123.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 46.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 547.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3T + 9.12e5T^{2} \) |
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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
−8.752893435434179807092589020598, −8.128492161959012685480570956475, −7.44063718675487939688833999677, −6.86185151681756448688125697595, −5.22649852598562012707837008044, −4.73261829764607522386804704492, −3.75565105801924994135118994312, −2.76578288630528469667021432288, −1.23412946772039298846437281409, 0,
1.23412946772039298846437281409, 2.76578288630528469667021432288, 3.75565105801924994135118994312, 4.73261829764607522386804704492, 5.22649852598562012707837008044, 6.86185151681756448688125697595, 7.44063718675487939688833999677, 8.128492161959012685480570956475, 8.752893435434179807092589020598
