L(s) = 1 | + 0.261·2-s + (−0.733 + 1.56i)3-s − 1.93·4-s − i·5-s + (−0.192 + 0.411i)6-s + i·7-s − 1.03·8-s + (−1.92 − 2.30i)9-s − 0.261i·10-s + (3.25 − 0.656i)11-s + (1.41 − 3.03i)12-s − 3.73i·13-s + 0.261i·14-s + (1.56 + 0.733i)15-s + 3.59·16-s + 1.46·17-s + ⋯ |
L(s) = 1 | + 0.185·2-s + (−0.423 + 0.905i)3-s − 0.965·4-s − 0.447i·5-s + (−0.0784 + 0.167i)6-s + 0.377i·7-s − 0.364·8-s + (−0.641 − 0.767i)9-s − 0.0828i·10-s + (0.980 − 0.197i)11-s + (0.408 − 0.874i)12-s − 1.03i·13-s + 0.0700i·14-s + (0.405 + 0.189i)15-s + 0.898·16-s + 0.355·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.081607183\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.081607183\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.733 - 1.56i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-3.25 + 0.656i)T \) |
good | 2 | \( 1 - 0.261T + 2T^{2} \) |
| 13 | \( 1 + 3.73iT - 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 - 5.82iT - 19T^{2} \) |
| 23 | \( 1 - 2.98iT - 23T^{2} \) |
| 29 | \( 1 + 4.36T + 29T^{2} \) |
| 31 | \( 1 - 5.20T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 + 3.91T + 41T^{2} \) |
| 43 | \( 1 - 1.80iT - 43T^{2} \) |
| 47 | \( 1 - 8.56iT - 47T^{2} \) |
| 53 | \( 1 - 14.0iT - 53T^{2} \) |
| 59 | \( 1 - 4.17iT - 59T^{2} \) |
| 61 | \( 1 - 0.658iT - 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 0.170iT - 71T^{2} \) |
| 73 | \( 1 - 7.57iT - 73T^{2} \) |
| 79 | \( 1 + 12.2iT - 79T^{2} \) |
| 83 | \( 1 - 5.56T + 83T^{2} \) |
| 89 | \( 1 - 7.47iT - 89T^{2} \) |
| 97 | \( 1 - 9.21T + 97T^{2} \) |
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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
−9.810615545034791200312370122890, −9.299282745260371487312403278704, −8.524208144580509779078003805661, −7.77995200721679183786658252693, −6.08003407480710767560980514686, −5.69270253519546835295659589090, −4.79028137394460294826130836947, −3.93437985589247336550999259130, −3.20955519632580439334312633958, −1.04766621326264608174187324307,
0.62612811563493405551366209018, 2.04659333388825678151871104317, 3.48206417168055652937475480153, 4.48299560648195882532466206598, 5.31384617770157188924719435719, 6.56117070939516072582297289158, 6.82761685546153454468374569754, 7.946792925455938999385507222264, 8.792481837334834870363615719358, 9.534821445870761468333515497114