L(s) = 1 | + (26.2 − 125. i)2-s − 1.26e3i·3-s + (−1.50e4 − 6.56e3i)4-s − 4.03e4·5-s + (−1.58e5 − 3.30e4i)6-s + 4.21e5i·7-s + (−1.21e6 + 1.70e6i)8-s − 1.59e6·9-s + (−1.05e6 + 5.05e6i)10-s − 1.42e6i·11-s + (−8.29e6 + 1.89e7i)12-s − 2.76e7·13-s + (5.27e7 + 1.10e7i)14-s + 5.09e7i·15-s + (1.82e8 + 1.97e8i)16-s + 3.56e8·17-s + ⋯ |
L(s) = 1 | + (0.204 − 0.978i)2-s − 0.577i·3-s + (−0.916 − 0.400i)4-s − 0.516·5-s + (−0.565 − 0.118i)6-s + 0.511i·7-s + (−0.579 + 0.814i)8-s − 0.333·9-s + (−0.105 + 0.505i)10-s − 0.0730i·11-s + (−0.231 + 0.528i)12-s − 0.440·13-s + (0.500 + 0.104i)14-s + 0.298i·15-s + (0.678 + 0.734i)16-s + 0.869·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(0.237076 + 0.155060i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.237076 + 0.155060i\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-26.2 + 125. i)T \) |
| 3 | \( 1 + 1.26e3iT \) |
good | 5 | \( 1 + 4.03e4T + 6.10e9T^{2} \) |
| 7 | \( 1 - 4.21e5iT - 6.78e11T^{2} \) |
| 11 | \( 1 + 1.42e6iT - 3.79e14T^{2} \) |
| 13 | \( 1 + 2.76e7T + 3.93e15T^{2} \) |
| 17 | \( 1 - 3.56e8T + 1.68e17T^{2} \) |
| 19 | \( 1 - 3.85e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 - 6.03e9iT - 1.15e19T^{2} \) |
| 29 | \( 1 + 1.53e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + 3.72e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 + 1.69e11T + 9.01e21T^{2} \) |
| 41 | \( 1 + 3.36e11T + 3.79e22T^{2} \) |
| 43 | \( 1 - 2.13e11iT - 7.38e22T^{2} \) |
| 47 | \( 1 - 4.76e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 1.64e12T + 1.37e24T^{2} \) |
| 59 | \( 1 - 9.85e10iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 2.21e12T + 9.87e24T^{2} \) |
| 67 | \( 1 + 9.25e12iT - 3.67e25T^{2} \) |
| 71 | \( 1 - 4.91e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 8.74e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + 5.31e12iT - 3.68e26T^{2} \) |
| 83 | \( 1 - 1.43e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 1.13e13T + 1.95e27T^{2} \) |
| 97 | \( 1 + 8.88e13T + 6.52e27T^{2} \) |
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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
−17.28373967819838260460741133163, −15.20816216075721727083902888738, −13.75555294264594146294013946515, −12.36167189875496425536426832502, −11.44717586725821263221950731315, −9.606539420062448703532770005093, −7.892426600815277929036813597723, −5.52149918128548123526339002437, −3.46843298766986930891094852079, −1.71572991322867274284609255670,
0.10729446479312400363926981633, 3.64486287061581745947291715005, 5.07277818730817403334009984411, 7.02079718892856582828974732004, 8.528694302200133656236478547976, 10.21439612473042673709401239695, 12.25003907623083319967241517338, 13.97162400303185740921288420897, 15.12238275879557536676420856448, 16.29477478796207104950966081556