| L(s) = 1 | − 1.56·2-s − 5.56·4-s + 7·7-s + 21.1·8-s + 10.2·11-s − 34.3·13-s − 10.9·14-s + 11.4·16-s − 82.6·17-s + 90.7·19-s − 16·22-s − 12.1·23-s + 53.6·26-s − 38.9·28-s + 105.·29-s − 142.·31-s − 187.·32-s + 128.·34-s − 64.8·37-s − 141.·38-s + 195.·41-s + 319.·43-s − 56.9·44-s + 18.9·46-s − 318.·47-s + 49·49-s + 191.·52-s + ⋯ |
| L(s) = 1 | − 0.552·2-s − 0.695·4-s + 0.377·7-s + 0.935·8-s + 0.280·11-s − 0.732·13-s − 0.208·14-s + 0.178·16-s − 1.17·17-s + 1.09·19-s − 0.155·22-s − 0.110·23-s + 0.404·26-s − 0.262·28-s + 0.673·29-s − 0.823·31-s − 1.03·32-s + 0.650·34-s − 0.288·37-s − 0.604·38-s + 0.743·41-s + 1.13·43-s − 0.195·44-s + 0.0607·46-s − 0.989·47-s + 0.142·49-s + 0.509·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 1.56T + 8T^{2} \) |
| 11 | \( 1 - 10.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 12.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 105.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 142.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 64.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 319.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 296.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 284T + 2.05e5T^{2} \) |
| 61 | \( 1 + 494.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 549.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 740.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 556.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 376.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 752.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 945.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 180.T + 9.12e5T^{2} \) |
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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
−8.868738706159550935696963911808, −7.86596886154450135500752780172, −7.35832138870063003989500672696, −6.29938672590132057834484863921, −5.14565052382689317365983629469, −4.59449175593685201203137836104, −3.59802994012841452569613876974, −2.27167902505665643537899197277, −1.12816888764171786518160287592, 0,
1.12816888764171786518160287592, 2.27167902505665643537899197277, 3.59802994012841452569613876974, 4.59449175593685201203137836104, 5.14565052382689317365983629469, 6.29938672590132057834484863921, 7.35832138870063003989500672696, 7.86596886154450135500752780172, 8.868738706159550935696963911808
