L(s) = 1 | + (−0.672 − 0.739i)5-s + (0.644 + 0.764i)7-s + (0.243 + 0.969i)11-s + (−0.584 − 0.811i)13-s + (0.988 + 0.150i)17-s + (−0.387 − 0.922i)19-s + (−0.726 − 0.686i)23-s + (−0.0944 + 0.995i)25-s + (0.726 − 0.686i)29-s + (−0.898 + 0.438i)31-s + (0.132 − 0.991i)35-s + (0.862 + 0.505i)37-s + (0.614 − 0.788i)41-s + (0.982 + 0.188i)43-s + (−0.999 + 0.0378i)47-s + ⋯ |
L(s) = 1 | + (−0.672 − 0.739i)5-s + (0.644 + 0.764i)7-s + (0.243 + 0.969i)11-s + (−0.584 − 0.811i)13-s + (0.988 + 0.150i)17-s + (−0.387 − 0.922i)19-s + (−0.726 − 0.686i)23-s + (−0.0944 + 0.995i)25-s + (0.726 − 0.686i)29-s + (−0.898 + 0.438i)31-s + (0.132 − 0.991i)35-s + (0.862 + 0.505i)37-s + (0.614 − 0.788i)41-s + (0.982 + 0.188i)43-s + (−0.999 + 0.0378i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.394166366 - 0.9839476926i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.394166366 - 0.9839476926i\) |
\(L(1)\) |
\(\approx\) |
\(0.9933539048 - 0.09359512865i\) |
\(L(1)\) |
\(\approx\) |
\(0.9933539048 - 0.09359512865i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 \) |
good | 5 | \( 1 + (-0.672 - 0.739i)T \) |
| 7 | \( 1 + (0.644 + 0.764i)T \) |
| 11 | \( 1 + (0.243 + 0.969i)T \) |
| 13 | \( 1 + (-0.584 - 0.811i)T \) |
| 17 | \( 1 + (0.988 + 0.150i)T \) |
| 19 | \( 1 + (-0.387 - 0.922i)T \) |
| 23 | \( 1 + (-0.726 - 0.686i)T \) |
| 29 | \( 1 + (0.726 - 0.686i)T \) |
| 31 | \( 1 + (-0.898 + 0.438i)T \) |
| 37 | \( 1 + (0.862 + 0.505i)T \) |
| 41 | \( 1 + (0.614 - 0.788i)T \) |
| 43 | \( 1 + (0.982 + 0.188i)T \) |
| 47 | \( 1 + (-0.999 + 0.0378i)T \) |
| 53 | \( 1 + (0.942 - 0.334i)T \) |
| 59 | \( 1 + (0.988 - 0.150i)T \) |
| 61 | \( 1 + (0.0189 + 0.999i)T \) |
| 67 | \( 1 + (-0.672 + 0.739i)T \) |
| 71 | \( 1 + (-0.974 + 0.225i)T \) |
| 73 | \( 1 + (-0.280 + 0.959i)T \) |
| 79 | \( 1 + (-0.752 - 0.658i)T \) |
| 83 | \( 1 + (-0.316 - 0.948i)T \) |
| 89 | \( 1 + (0.843 - 0.537i)T \) |
| 97 | \( 1 + (0.898 + 0.438i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.3786282015091676492788139010, −17.96022034945867035254363919740, −16.85529589368349458480780261083, −16.50980537744114247474022061776, −15.86519295612171964864990371821, −14.67590737869587391508650956722, −14.45432884064401026918732520343, −13.97640992226593867961789989974, −12.99534092341343187862612810863, −11.999100530538363091137399682417, −11.60317004958188243935059399965, −10.893762550177182110925057384596, −10.30063165042131650312693185419, −9.521458270422519151274860115390, −8.5637147928851992318238522467, −7.68674515729610578583251277576, −7.53495354288091001723169969676, −6.50094625806000826647524980673, −5.84228943565968507956304880441, −4.8410168637722333795005815946, −3.93812881038663584435750531514, −3.60187632838141619169201982329, −2.577844234102099521656279586542, −1.58815110374204374417702593050, −0.69812107669990265419696164699,
0.35085756764911413361867535116, 1.23776994265672142837659135341, 2.206988423232332502572966464288, 2.91648478982072888839078787221, 4.116915712147963056277777623658, 4.60811621931748482627858313070, 5.33270369232294820649451895912, 6.00631261922763480720205616652, 7.2063273778142008846229369660, 7.7015166177493806162458405694, 8.44872633479722869528460429751, 8.988913049928573504936137517, 9.88045253806054342354598468434, 10.51225309678380590976011726246, 11.67084848378494996729470820186, 11.8766145569237127834542390775, 12.73076755477597683758438730097, 13.03457937147114190502660092539, 14.49121652742665207765339643964, 14.67576450763685912194947147041, 15.48884739648286621295456157216, 16.03152650268128491989616760712, 16.84873463643701653978380306279, 17.64327925674099282831589267854, 17.93576070795625545374653023630