L(s) = 1 | − 1.39·2-s + 3-s − 0.0663·4-s − 1.39·6-s + 4.73·7-s + 2.87·8-s + 9-s − 0.0663·12-s + 3.26·13-s − 6.57·14-s − 3.86·16-s + 3.79·17-s − 1.39·18-s − 1.59·19-s + 4.73·21-s − 0.467·23-s + 2.87·24-s − 4.53·26-s + 27-s − 0.313·28-s − 4.44·29-s − 9.29·31-s − 0.375·32-s − 5.27·34-s − 0.0663·36-s + 0.117·37-s + 2.22·38-s + ⋯ |
L(s) = 1 | − 0.983·2-s + 0.577·3-s − 0.0331·4-s − 0.567·6-s + 1.78·7-s + 1.01·8-s + 0.333·9-s − 0.0191·12-s + 0.905·13-s − 1.75·14-s − 0.965·16-s + 0.920·17-s − 0.327·18-s − 0.366·19-s + 1.03·21-s − 0.0974·23-s + 0.586·24-s − 0.890·26-s + 0.192·27-s − 0.0592·28-s − 0.825·29-s − 1.66·31-s − 0.0663·32-s − 0.905·34-s − 0.0110·36-s + 0.0192·37-s + 0.360·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.977692031\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.977692031\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.39T + 2T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 + 1.59T + 19T^{2} \) |
| 23 | \( 1 + 0.467T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 + 9.29T + 31T^{2} \) |
| 37 | \( 1 - 0.117T + 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 - 3.44T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 5.92T + 53T^{2} \) |
| 59 | \( 1 - 4.23T + 59T^{2} \) |
| 61 | \( 1 + 3.54T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 6.72T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 - 5.57T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.899178115914752898950809489399, −7.52490043335145113473631749562, −6.63949269439506927176617561593, −5.43369701953489738135804757484, −5.08532296654345153874368247028, −4.05989146853522613043905640199, −3.63023153564783794432936958554, −2.15690676070540355259723741427, −1.65642401162846244015253612426, −0.833304190565447670878538308239,
0.833304190565447670878538308239, 1.65642401162846244015253612426, 2.15690676070540355259723741427, 3.63023153564783794432936958554, 4.05989146853522613043905640199, 5.08532296654345153874368247028, 5.43369701953489738135804757484, 6.63949269439506927176617561593, 7.52490043335145113473631749562, 7.899178115914752898950809489399