| L(s) = 1 | + 0.618i·2-s + 0.381i·3-s + 1.61·4-s − 0.236·6-s − 3.85i·7-s + 2.23i·8-s + 2.85·9-s + 11-s + 0.618i·12-s + 1.76i·13-s + 2.38·14-s + 1.85·16-s − 1.61i·17-s + 1.76i·18-s − 6.70·19-s + ⋯ |
| L(s) = 1 | + 0.437i·2-s + 0.220i·3-s + 0.809·4-s − 0.0963·6-s − 1.45i·7-s + 0.790i·8-s + 0.951·9-s + 0.301·11-s + 0.178i·12-s + 0.489i·13-s + 0.636·14-s + 0.463·16-s − 0.392i·17-s + 0.415i·18-s − 1.53·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55421 + 0.366900i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55421 + 0.366900i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618iT - 2T^{2} \) |
| 3 | \( 1 - 0.381iT - 3T^{2} \) |
| 7 | \( 1 + 3.85iT - 7T^{2} \) |
| 13 | \( 1 - 1.76iT - 13T^{2} \) |
| 17 | \( 1 + 1.61iT - 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 - 7.09iT - 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 5.76iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 5.94iT - 47T^{2} \) |
| 53 | \( 1 + 6.32iT - 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 0.854T + 79T^{2} \) |
| 83 | \( 1 + 16.8iT - 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 - 0.618iT - 97T^{2} \) |
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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
−11.84461710480034298367198611132, −10.83094550040838967459368947322, −10.30987805828182031794443996208, −9.126664644072682482411526478037, −7.67702977212959500439482088540, −7.12491703748878456259843067434, −6.25007262721784143313310902557, −4.68428314718568276906684824242, −3.65291961015538061582749898950, −1.69333495200440983818429169325,
1.78884995383060609452971727447, 2.86640126537878796844429070088, 4.47096473073473755062276571876, 6.05815394170619843018105577068, 6.67490508805231190271068237970, 8.017247639143871419598944526127, 8.962153741404587583247385488691, 10.18503816034326055926976106222, 10.84134021148139915404776211578, 12.13066480227664940063139510734
