| L(s) = 1 | + (23.4 + 13.5i)5-s + (−2.30e3 − 663. i)7-s + (−6.53e3 + 3.77e3i)11-s + 1.07e4·13-s + (4.87e4 − 2.81e4i)17-s + (5.54e4 − 9.60e4i)19-s + (−4.09e5 − 2.36e5i)23-s + (−1.94e5 − 3.37e5i)25-s + 3.86e5i·29-s + (7.68e5 + 1.33e6i)31-s + (−4.50e4 − 4.67e4i)35-s + (7.92e5 − 1.37e6i)37-s − 7.94e5i·41-s − 6.19e6·43-s + (6.61e6 + 3.81e6i)47-s + ⋯ |
| L(s) = 1 | + (0.0374 + 0.0216i)5-s + (−0.961 − 0.276i)7-s + (−0.446 + 0.257i)11-s + 0.375·13-s + (0.584 − 0.337i)17-s + (0.425 − 0.737i)19-s + (−1.46 − 0.845i)23-s + (−0.499 − 0.864i)25-s + 0.545i·29-s + (0.831 + 1.44i)31-s + (−0.0300 − 0.0311i)35-s + (0.423 − 0.732i)37-s − 0.281i·41-s − 1.81·43-s + (1.35 + 0.782i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.9838735812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9838735812\) |
| \(L(5)\) |
|
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 | \( 1 + (2.30e3 + 663. i)T \) |
| good | 5 | \( 1 + (-23.4 - 13.5i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (6.53e3 - 3.77e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 1.07e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-4.87e4 + 2.81e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-5.54e4 + 9.60e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (4.09e5 + 2.36e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 3.86e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-7.68e5 - 1.33e6i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-7.92e5 + 1.37e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 7.94e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 6.19e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-6.61e6 - 3.81e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-2.25e6 + 1.30e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.13e7 - 6.54e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (2.09e6 - 3.62e6i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (3.05e6 + 5.28e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.56e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (-4.61e5 - 7.99e5i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (9.41e5 - 1.63e6i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 8.96e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (2.07e7 + 1.19e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 1.05e7T + 7.83e15T^{2} \) |
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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
−10.57476404561349254339579872854, −10.01817270569297728773232964410, −8.973865952194350626347647688979, −7.87720135369059768350591607797, −6.83318959519099270362279996410, −5.94077659975972677959938081645, −4.67526307425357512349298678579, −3.47298402265144507029094096873, −2.42462866310254060133353861425, −0.851353734313436543831646280158,
0.26323136958064232271418200834, 1.72278467917171861804176706455, 3.07927694614236898863109477294, 3.98295108111033473163724210483, 5.61551860802740628920627163716, 6.16415220467430818155528999904, 7.55555267410890178333264629381, 8.375934725127591820982719661357, 9.710708144919177982120566011712, 10.08105071995559594330893515310
