L(s) = 1 | + 16.9i·11-s + 2·17-s + 16.9i·19-s − 25·25-s − 46·41-s − 84.8i·43-s + 49·49-s + 84.8i·59-s − 118. i·67-s − 142·73-s + 50.9i·83-s − 146·89-s + 94·97-s − 118. i·107-s − 98·113-s + ⋯ |
L(s) = 1 | + 1.54i·11-s + 0.117·17-s + 0.893i·19-s − 25-s − 1.12·41-s − 1.97i·43-s + 0.999·49-s + 1.43i·59-s − 1.77i·67-s − 1.94·73-s + 0.613i·83-s − 1.64·89-s + 0.969·97-s − 1.11i·107-s − 0.867·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4471725617\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4471725617\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 16.9iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 2T + 289T^{2} \) |
| 19 | \( 1 - 16.9iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 46T + 1.68e3T^{2} \) |
| 43 | \( 1 + 84.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 84.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 + 118. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 142T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 50.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 146T + 7.92e3T^{2} \) |
| 97 | \( 1 - 94T + 9.40e3T^{2} \) |
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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
−9.235384836471444207042729171731, −8.448139312718871238150161127569, −7.52720701895520787846921348283, −7.08125241648226899241500919212, −6.05503360488972100009877396828, −5.27411531435400442767236022139, −4.35655484238531751557438516364, −3.62108881631408503189877016112, −2.33839353386164092967432426258, −1.53426115256905361727109725381,
0.10814679148167641565044134418, 1.27201985630527599185413888042, 2.63214251248987863495303488788, 3.41815580981282707536706100955, 4.37285108708539719384653675937, 5.38520497966722607213080345193, 6.05175578867713213553138427863, 6.83616088820026541030727861164, 7.78625039234257766624864220524, 8.454812691377401612463180879601