| L(s) = 1 | − 0.548i·2-s + (1.56 − 0.745i)3-s + 1.69·4-s − 2.39·5-s + (−0.408 − 0.857i)6-s − i·7-s − 2.02i·8-s + (1.88 − 2.33i)9-s + 1.31i·10-s + 2.76·11-s + (2.65 − 1.26i)12-s − 4.35·13-s − 0.548·14-s + (−3.74 + 1.78i)15-s + 2.28·16-s + 1.94·17-s + ⋯ |
| L(s) = 1 | − 0.387i·2-s + (0.902 − 0.430i)3-s + 0.849·4-s − 1.07·5-s + (−0.166 − 0.350i)6-s − 0.377i·7-s − 0.717i·8-s + (0.629 − 0.776i)9-s + 0.415i·10-s + 0.832·11-s + (0.766 − 0.365i)12-s − 1.20·13-s − 0.146·14-s + (−0.966 + 0.460i)15-s + 0.571·16-s + 0.472·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.51236 - 1.27980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51236 - 1.27980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.56 + 0.745i)T \) |
| 7 | \( 1 + iT \) |
| 23 | \( 1 + (-4.61 - 1.31i)T \) |
| good | 2 | \( 1 + 0.548iT - 2T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 11 | \( 1 - 2.76T + 11T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 + 6.19iT - 19T^{2} \) |
| 29 | \( 1 - 8.98iT - 29T^{2} \) |
| 31 | \( 1 + 0.0448T + 31T^{2} \) |
| 37 | \( 1 - 5.35iT - 37T^{2} \) |
| 41 | \( 1 - 2.74iT - 41T^{2} \) |
| 43 | \( 1 - 5.93iT - 43T^{2} \) |
| 47 | \( 1 + 1.71iT - 47T^{2} \) |
| 53 | \( 1 + 6.76T + 53T^{2} \) |
| 59 | \( 1 - 0.301iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 4.99iT - 67T^{2} \) |
| 71 | \( 1 + 5.49iT - 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 5.43iT - 79T^{2} \) |
| 83 | \( 1 + 2.33T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 8.26iT - 97T^{2} \) |
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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
−10.98641443819326006930132774584, −9.847540605641773271129303080637, −9.023098258553546189282019280382, −7.86723940284595270700626937110, −7.17636560322263517611842537796, −6.67883756991159643372300582737, −4.73241925982778960740896194672, −3.53513825267556700477554590154, −2.77277647771753389676635729882, −1.21611534043614029307241348227,
2.07576608347292884669660947205, 3.28433644752833965511833070687, 4.26607720917593792259118403999, 5.55510654384660302694001585727, 6.86366560459081160381959923758, 7.71734761814244117187746879890, 8.182485796101973282710352320865, 9.338282471135233136840150849531, 10.19613226911584500768289122760, 11.25278897700083867729615035288
