L(s) = 1 | − 1.04·3-s − 3.77·5-s + 1.06·7-s − 1.91·9-s + 3.78·11-s − 4.91·13-s + 3.93·15-s + 3.62·17-s − 6.79·19-s − 1.11·21-s − 6.61·23-s + 9.25·25-s + 5.12·27-s − 0.497·29-s − 6.78·31-s − 3.94·33-s − 4.02·35-s − 4.78·37-s + 5.12·39-s + 10.4·41-s + 1.82·43-s + 7.22·45-s − 6.43·47-s − 5.86·49-s − 3.77·51-s − 11.9·53-s − 14.2·55-s + ⋯ |
L(s) = 1 | − 0.601·3-s − 1.68·5-s + 0.402·7-s − 0.637·9-s + 1.14·11-s − 1.36·13-s + 1.01·15-s + 0.879·17-s − 1.55·19-s − 0.242·21-s − 1.37·23-s + 1.85·25-s + 0.985·27-s − 0.0924·29-s − 1.21·31-s − 0.686·33-s − 0.680·35-s − 0.786·37-s + 0.820·39-s + 1.63·41-s + 0.277·43-s + 1.07·45-s − 0.938·47-s − 0.837·49-s − 0.529·51-s − 1.64·53-s − 1.92·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3592006721\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3592006721\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 + 1.04T + 3T^{2} \) |
| 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 - 1.06T + 7T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 + 4.91T + 13T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 19 | \( 1 + 6.79T + 19T^{2} \) |
| 23 | \( 1 + 6.61T + 23T^{2} \) |
| 29 | \( 1 + 0.497T + 29T^{2} \) |
| 31 | \( 1 + 6.78T + 31T^{2} \) |
| 37 | \( 1 + 4.78T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 + 6.43T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 1.88T + 59T^{2} \) |
| 61 | \( 1 - 7.47T + 61T^{2} \) |
| 67 | \( 1 + 0.911T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 3.44T + 79T^{2} \) |
| 83 | \( 1 - 3.28T + 83T^{2} \) |
| 89 | \( 1 - 2.12T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−7.944339994079647617060326157357, −7.51186978758653416686142887348, −6.68944942168721302593898379476, −5.99800639303088872401775631472, −5.09316215187021221788142770310, −4.36301617300333586648415115234, −3.85908686862994399766242756875, −2.93483236902751493549036039656, −1.73791957755607757532186214528, −0.31718937926131403401122517032,
0.31718937926131403401122517032, 1.73791957755607757532186214528, 2.93483236902751493549036039656, 3.85908686862994399766242756875, 4.36301617300333586648415115234, 5.09316215187021221788142770310, 5.99800639303088872401775631472, 6.68944942168721302593898379476, 7.51186978758653416686142887348, 7.944339994079647617060326157357