L(s) = 1 | + 0.430i·2-s − 3-s + 1.81·4-s + 1.89i·5-s − 0.430i·6-s − 3.40i·7-s + 1.64i·8-s + 9-s − 0.815·10-s + 2.33·11-s − 1.81·12-s + 1.46·13-s + 1.46·14-s − 1.89i·15-s + 2.92·16-s + 5.58·17-s + ⋯ |
L(s) = 1 | + 0.304i·2-s − 0.577·3-s + 0.907·4-s + 0.847i·5-s − 0.175i·6-s − 1.28i·7-s + 0.580i·8-s + 0.333·9-s − 0.257·10-s + 0.703·11-s − 0.523·12-s + 0.405·13-s + 0.391·14-s − 0.489i·15-s + 0.731·16-s + 1.35·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48691 + 0.455687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48691 + 0.455687i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 157 | \( 1 + (10.3 - 7.02i)T \) |
good | 2 | \( 1 - 0.430iT - 2T^{2} \) |
| 5 | \( 1 - 1.89iT - 5T^{2} \) |
| 7 | \( 1 + 3.40iT - 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 - 5.58T + 17T^{2} \) |
| 19 | \( 1 + 6.04T + 19T^{2} \) |
| 23 | \( 1 - 8.56iT - 23T^{2} \) |
| 29 | \( 1 + 8.19iT - 29T^{2} \) |
| 31 | \( 1 + 4.52T + 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 - 3.26iT - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 1.67T + 47T^{2} \) |
| 53 | \( 1 + 0.511iT - 53T^{2} \) |
| 59 | \( 1 + 9.95iT - 59T^{2} \) |
| 61 | \( 1 + 1.38iT - 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 16.3iT - 73T^{2} \) |
| 79 | \( 1 + 1.85iT - 79T^{2} \) |
| 83 | \( 1 - 3.55iT - 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 + 16.7iT - 97T^{2} \) |
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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
−11.17368705767769496743363180133, −10.40982947977186338944384632390, −9.639575116421641971697370307612, −7.88734337678687938745714547762, −7.37599766364338137776876537679, −6.43827123617093046638420519679, −5.88419405216820050661670081827, −4.24811210790501722232640929321, −3.19023641857234563727825434211, −1.44104940095751852713150830469,
1.29491582112469339467868336614, 2.61776995217320834797187931268, 4.12508580286534252360239450144, 5.44917025251564198018089750923, 6.11780822630854692766725200942, 7.08176709351400347881411599800, 8.478610308636150788681484704470, 9.031679013832578378579704961070, 10.29915603375029664160725912498, 10.98121403186153826754601357417