L(s) = 1 | + (1.33e3 − 2.30e3i)5-s + (4.36e4 − 8.49e3i)7-s + (−2.62e5 − 4.55e5i)11-s − 1.10e6·13-s + (2.22e5 + 3.84e5i)17-s + (−1.58e6 + 2.74e6i)19-s + (−2.38e7 + 4.13e7i)23-s + (2.08e7 + 3.61e7i)25-s + 1.65e8·29-s + (−8.05e7 − 1.39e8i)31-s + (3.85e7 − 1.12e8i)35-s + (−1.73e8 + 3.00e8i)37-s + 1.07e9·41-s + 1.10e9·43-s + (4.49e7 − 7.77e7i)47-s + ⋯ |
L(s) = 1 | + (0.190 − 0.330i)5-s + (0.981 − 0.191i)7-s + (−0.492 − 0.852i)11-s − 0.823·13-s + (0.0379 + 0.0656i)17-s + (−0.146 + 0.254i)19-s + (−0.773 + 1.34i)23-s + (0.427 + 0.740i)25-s + 1.49·29-s + (−0.505 − 0.875i)31-s + (0.124 − 0.360i)35-s + (−0.410 + 0.711i)37-s + 1.44·41-s + 1.14·43-s + (0.0285 − 0.0494i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(2.428157201\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.428157201\) |
\(L(\frac{13}{2})\) |
|
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 | \( 1 + (-4.36e4 + 8.49e3i)T \) |
good | 5 | \( 1 + (-1.33e3 + 2.30e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (2.62e5 + 4.55e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.10e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-2.22e5 - 3.84e5i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (1.58e6 - 2.74e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (2.38e7 - 4.13e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.65e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (8.05e7 + 1.39e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (1.73e8 - 3.00e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 1.07e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.10e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-4.49e7 + 7.77e7i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (1.97e8 + 3.42e8i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (3.65e9 + 6.32e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (6.05e9 - 1.04e10i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-8.63e9 - 1.49e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.31e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (7.39e9 + 1.28e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (1.45e10 - 2.51e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 2.43e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-4.46e10 + 7.73e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 6.63e10T + 7.15e21T^{2} \) |
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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
−10.10035748784713784947047474132, −9.062128623177926129350447519699, −8.071633503419484379937240693857, −7.39553144509912631685194372303, −5.92449904542143900171839539670, −5.14197302344537922759087301800, −4.12793397151430034353632144130, −2.80700407580342462358258891259, −1.66191399229931823655731984680, −0.66579353185863972853120481445,
0.65263963380296872451600853685, 2.05766489806816605823199505451, 2.66503789779317326907691068259, 4.38598306761671867860699867819, 4.98741914269369205671715759508, 6.26880100869699539910043183514, 7.31990932757842983325582698990, 8.145504300783828793318440676195, 9.187626930934932557588770374013, 10.34115707998617110572060386270