L(s) = 1 | + (−173. − 299. i)5-s + (334. + 843. i)7-s + (−437. + 758. i)11-s − 507.·13-s + (9.80e3 − 1.69e4i)17-s + (1.77e4 + 3.07e4i)19-s + (−4.87e4 − 8.44e4i)23-s + (−2.08e4 + 3.61e4i)25-s − 5.66e4·29-s + (2.63e4 − 4.56e4i)31-s + (1.95e5 − 2.46e5i)35-s + (−1.71e5 − 2.97e5i)37-s − 2.79e5·41-s + 7.37e4·43-s + (5.14e5 + 8.91e5i)47-s + ⋯ |
L(s) = 1 | + (−0.619 − 1.07i)5-s + (0.368 + 0.929i)7-s + (−0.0991 + 0.171i)11-s − 0.0640·13-s + (0.483 − 0.838i)17-s + (0.594 + 1.02i)19-s + (−0.835 − 1.44i)23-s + (−0.267 + 0.463i)25-s − 0.431·29-s + (0.158 − 0.274i)31-s + (0.769 − 0.970i)35-s + (−0.558 − 0.966i)37-s − 0.633·41-s + 0.141·43-s + (0.723 + 1.25i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.3170128839\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3170128839\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-334. - 843. i)T \) |
good | 5 | \( 1 + (173. + 299. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (437. - 758. i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + 507.T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-9.80e3 + 1.69e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.77e4 - 3.07e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (4.87e4 + 8.44e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + 5.66e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-2.63e4 + 4.56e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.71e5 + 2.97e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + 2.79e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.37e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-5.14e5 - 8.91e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (1.86e5 - 3.23e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (5.20e5 - 9.01e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.61e5 - 2.79e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.50e6 - 2.60e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 2.97e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (1.26e6 - 2.19e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-9.36e5 - 1.62e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + 3.44e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (5.03e6 + 8.71e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.52e7T + 8.07e13T^{2} \) |
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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
−11.41066554561181924580494550224, −10.09964219390395515598912242011, −9.073554508409966954467580873758, −8.309582241130931347954621769001, −7.49836745461527713403653919824, −5.92231538218799042843878169253, −5.04235193916409416342815077282, −4.05388599431701583853179680927, −2.53893700410287288746494213731, −1.19370101748091007767074527603,
0.07796492368229624761206507996, 1.55093840107216416976266542117, 3.16118571224497916320979507891, 3.91209330912792132534865203931, 5.28840830105591467061769513879, 6.68057142722241790176330562035, 7.43279171260964276469615026704, 8.209017747820723942122358064467, 9.679954961890300835068462501585, 10.58356121693807119256740189972