L(s) = 1 | + (−62.3 + 14.2i)2-s + 420. i·3-s + (3.68e3 − 1.78e3i)4-s − 2.16e4·5-s + (−6.00e3 − 2.62e4i)6-s + 1.55e5i·7-s + (−2.04e5 + 1.63e5i)8-s − 1.77e5·9-s + (1.34e6 − 3.08e5i)10-s − 3.12e6i·11-s + (7.49e5 + 1.55e6i)12-s + 1.10e6·13-s + (−2.21e6 − 9.68e6i)14-s − 9.10e6i·15-s + (1.04e7 − 1.31e7i)16-s + 3.10e7·17-s + ⋯ |
L(s) = 1 | + (−0.974 + 0.223i)2-s + 0.577i·3-s + (0.900 − 0.434i)4-s − 1.38·5-s + (−0.128 − 0.562i)6-s + 1.31i·7-s + (−0.780 + 0.624i)8-s − 0.333·9-s + (1.34 − 0.308i)10-s − 1.76i·11-s + (0.251 + 0.519i)12-s + 0.228·13-s + (−0.294 − 1.28i)14-s − 0.798i·15-s + (0.621 − 0.783i)16-s + 1.28·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(0.380601 - 0.238876i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.380601 - 0.238876i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (62.3 - 14.2i)T \) |
| 3 | \( 1 - 420. iT \) |
good | 5 | \( 1 + 2.16e4T + 2.44e8T^{2} \) |
| 7 | \( 1 - 1.55e5iT - 1.38e10T^{2} \) |
| 11 | \( 1 + 3.12e6iT - 3.13e12T^{2} \) |
| 13 | \( 1 - 1.10e6T + 2.32e13T^{2} \) |
| 17 | \( 1 - 3.10e7T + 5.82e14T^{2} \) |
| 19 | \( 1 + 2.82e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + 1.30e8iT - 2.19e16T^{2} \) |
| 29 | \( 1 + 7.45e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + 3.42e8iT - 7.87e17T^{2} \) |
| 37 | \( 1 - 1.27e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + 8.57e8T + 2.25e19T^{2} \) |
| 43 | \( 1 + 6.63e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 - 7.75e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 + 1.80e10T + 4.91e20T^{2} \) |
| 59 | \( 1 - 6.86e9iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 2.30e10T + 2.65e21T^{2} \) |
| 67 | \( 1 + 1.09e11iT - 8.18e21T^{2} \) |
| 71 | \( 1 + 2.98e10iT - 1.64e22T^{2} \) |
| 73 | \( 1 + 2.57e11T + 2.29e22T^{2} \) |
| 79 | \( 1 + 3.42e11iT - 5.90e22T^{2} \) |
| 83 | \( 1 - 5.68e10iT - 1.06e23T^{2} \) |
| 89 | \( 1 - 6.63e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + 9.94e11T + 6.93e23T^{2} \) |
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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
−16.58335511848992166062182188313, −15.81009171349001517876688051231, −14.78941583657195403525419376066, −11.92560223632206823300099073158, −10.99904865884171900354238807304, −9.004460555629182822207724539342, −8.027382968078493043183658916284, −5.77736618512218812365725679403, −3.15573045392254991552928718793, −0.31369821404862009919089008355,
1.28563939032096444202828659286, 3.75765091641612526520270160705, 7.24390834351371214833901658936, 7.78468345145206739555293990503, 9.983231475262399082878839932350, 11.50336258293104047644766423690, 12.64985482602895985804865068815, 14.88118756142163747599890839448, 16.36290849317211234855155665206, 17.51803908753072953952602099408