L(s) = 1 | + 2-s + 2.49·3-s + 4-s + 5-s + 2.49·6-s + 2.61·7-s + 8-s + 3.21·9-s + 10-s + 11-s + 2.49·12-s − 1.45·13-s + 2.61·14-s + 2.49·15-s + 16-s + 2.28·17-s + 3.21·18-s + 7.37·19-s + 20-s + 6.53·21-s + 22-s + 1.38·23-s + 2.49·24-s + 25-s − 1.45·26-s + 0.546·27-s + 2.61·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.43·3-s + 0.5·4-s + 0.447·5-s + 1.01·6-s + 0.989·7-s + 0.353·8-s + 1.07·9-s + 0.316·10-s + 0.301·11-s + 0.719·12-s − 0.403·13-s + 0.699·14-s + 0.643·15-s + 0.250·16-s + 0.553·17-s + 0.758·18-s + 1.69·19-s + 0.223·20-s + 1.42·21-s + 0.213·22-s + 0.288·23-s + 0.509·24-s + 0.200·25-s − 0.285·26-s + 0.105·27-s + 0.494·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.580492273\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.580492273\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 - 2.49T + 3T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 - 2.28T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 + 6.59T + 29T^{2} \) |
| 31 | \( 1 - 0.701T + 31T^{2} \) |
| 37 | \( 1 + 4.99T + 37T^{2} \) |
| 41 | \( 1 + 2.35T + 41T^{2} \) |
| 43 | \( 1 - 7.04T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 + 9.09T + 53T^{2} \) |
| 59 | \( 1 + 9.20T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 - 7.41T + 67T^{2} \) |
| 71 | \( 1 + 1.62T + 71T^{2} \) |
| 79 | \( 1 - 8.69T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 7.40T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.61159195510106062693362958385, −7.49233515006992514343178604053, −6.49458154780149892201362179507, −5.44506059194954884428042890161, −5.10623515617493052942680767363, −4.13865326694763412968354519517, −3.40553262219188504051222755946, −2.82263380662882701882453874811, −1.93008254707446752774537351153, −1.31609235316905972731541005414,
1.31609235316905972731541005414, 1.93008254707446752774537351153, 2.82263380662882701882453874811, 3.40553262219188504051222755946, 4.13865326694763412968354519517, 5.10623515617493052942680767363, 5.44506059194954884428042890161, 6.49458154780149892201362179507, 7.49233515006992514343178604053, 7.61159195510106062693362958385