L(s) = 1 | + 1.80·2-s + 1.25·4-s − 2.77·5-s + 7-s − 1.33·8-s − 5.01·10-s − 2.31·13-s + 1.80·14-s − 4.93·16-s + 2.66·17-s − 8.08·19-s − 3.49·20-s + 2.43·23-s + 2.71·25-s − 4.17·26-s + 1.25·28-s − 7.55·29-s + 9.24·31-s − 6.23·32-s + 4.80·34-s − 2.77·35-s + 11.1·37-s − 14.5·38-s + 3.71·40-s − 0.299·41-s + 7.29·43-s + 4.38·46-s + ⋯ |
L(s) = 1 | + 1.27·2-s + 0.629·4-s − 1.24·5-s + 0.377·7-s − 0.472·8-s − 1.58·10-s − 0.641·13-s + 0.482·14-s − 1.23·16-s + 0.645·17-s − 1.85·19-s − 0.782·20-s + 0.506·23-s + 0.542·25-s − 0.819·26-s + 0.238·28-s − 1.40·29-s + 1.66·31-s − 1.10·32-s + 0.824·34-s − 0.469·35-s + 1.84·37-s − 2.36·38-s + 0.586·40-s − 0.0468·41-s + 1.11·43-s + 0.646·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.212120541\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.212120541\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 - 2.66T + 17T^{2} \) |
| 19 | \( 1 + 8.08T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 7.55T + 29T^{2} \) |
| 31 | \( 1 - 9.24T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 0.299T + 41T^{2} \) |
| 43 | \( 1 - 7.29T + 43T^{2} \) |
| 47 | \( 1 + 0.457T + 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 1.81T + 61T^{2} \) |
| 67 | \( 1 + 1.42T + 67T^{2} \) |
| 71 | \( 1 + 0.558T + 71T^{2} \) |
| 73 | \( 1 - 9.66T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 - 1.56T + 89T^{2} \) |
| 97 | \( 1 + 0.533T + 97T^{2} \) |
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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
−7.80583616518033909102709544076, −7.12900923168911342966838071312, −6.29478016743337221925177835049, −5.70782290488733776496366539078, −4.71902378739701140950955025629, −4.39622578016523052636793643506, −3.78968112723148882206971895310, −2.94122104989068526483613529609, −2.17363011524403984393700531652, −0.58767375107835252409120330919,
0.58767375107835252409120330919, 2.17363011524403984393700531652, 2.94122104989068526483613529609, 3.78968112723148882206971895310, 4.39622578016523052636793643506, 4.71902378739701140950955025629, 5.70782290488733776496366539078, 6.29478016743337221925177835049, 7.12900923168911342966838071312, 7.80583616518033909102709544076