L(s) = 1 | + (−0.0747 + 0.997i)4-s + (−0.955 − 0.294i)7-s + (0.202 + 0.218i)13-s + (−0.988 − 0.149i)16-s + (1.68 + 0.974i)19-s + (−0.222 + 0.974i)25-s + (0.365 − 0.930i)28-s + (−0.975 − 0.563i)31-s + (−1.36 + 0.930i)37-s + (0.535 + 1.36i)43-s + (0.826 + 0.563i)49-s + (−0.233 + 0.185i)52-s + (−1.98 + 0.149i)61-s + (0.222 − 0.974i)64-s + (−0.365 + 0.632i)67-s + ⋯ |
L(s) = 1 | + (−0.0747 + 0.997i)4-s + (−0.955 − 0.294i)7-s + (0.202 + 0.218i)13-s + (−0.988 − 0.149i)16-s + (1.68 + 0.974i)19-s + (−0.222 + 0.974i)25-s + (0.365 − 0.930i)28-s + (−0.975 − 0.563i)31-s + (−1.36 + 0.930i)37-s + (0.535 + 1.36i)43-s + (0.826 + 0.563i)49-s + (−0.233 + 0.185i)52-s + (−1.98 + 0.149i)61-s + (0.222 − 0.974i)64-s + (−0.365 + 0.632i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8586878070\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8586878070\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.955 + 0.294i)T \) |
good | 2 | \( 1 + (0.0747 - 0.997i)T^{2} \) |
| 5 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (-0.202 - 0.218i)T + (-0.0747 + 0.997i)T^{2} \) |
| 17 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 19 | \( 1 + (-1.68 - 0.974i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 29 | \( 1 + (0.365 - 0.930i)T^{2} \) |
| 31 | \( 1 + (0.975 + 0.563i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.36 - 0.930i)T + (0.365 - 0.930i)T^{2} \) |
| 41 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 43 | \( 1 + (-0.535 - 1.36i)T + (-0.733 + 0.680i)T^{2} \) |
| 47 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 53 | \( 1 + (0.365 + 0.930i)T^{2} \) |
| 59 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 61 | \( 1 + (1.98 - 0.149i)T + (0.988 - 0.149i)T^{2} \) |
| 67 | \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.255 - 0.829i)T + (-0.826 + 0.563i)T^{2} \) |
| 79 | \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 89 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 97 | \( 1 + (-1.17 - 0.680i)T + (0.5 + 0.866i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.022374164217805751941205946052, −7.947570689903981180235446186482, −7.50687404737827032531848484927, −6.83225348530948740477983314727, −5.98746592638345907415110483098, −5.15535094388898651561127360625, −4.06802637434885839041657117167, −3.44812596508248330712675424465, −2.87684814488145382697501710325, −1.47600724528512333738501594933,
0.48548408706461466005693011345, 1.80804860237661146562377715402, 2.86704678295177389637665995430, 3.68551371530232781036469534303, 4.80895434787952981075929125780, 5.50171851986667669116617821780, 6.05326887505621461120306893310, 6.91953187249337585411733709237, 7.42487013737595577094152894544, 8.662903546883766706285106917482