L(s) = 1 | + 2.32·2-s + (−1.66 + 0.462i)3-s + 3.40·4-s + (0.5 − 0.866i)5-s + (−3.88 + 1.07i)6-s + (2.02 − 1.70i)7-s + 3.27·8-s + (2.57 − 1.54i)9-s + (1.16 − 2.01i)10-s + (1.47 + 2.56i)11-s + (−5.68 + 1.57i)12-s + (1.00 + 1.73i)13-s + (4.71 − 3.95i)14-s + (−0.434 + 1.67i)15-s + 0.799·16-s + (−1.98 + 3.43i)17-s + ⋯ |
L(s) = 1 | + 1.64·2-s + (−0.963 + 0.266i)3-s + 1.70·4-s + (0.223 − 0.387i)5-s + (−1.58 + 0.438i)6-s + (0.766 − 0.642i)7-s + 1.15·8-s + (0.857 − 0.514i)9-s + (0.367 − 0.636i)10-s + (0.445 + 0.772i)11-s + (−1.64 + 0.454i)12-s + (0.278 + 0.482i)13-s + (1.25 − 1.05i)14-s + (−0.112 + 0.432i)15-s + 0.199·16-s + (−0.480 + 0.831i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.58091 - 0.0603122i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.58091 - 0.0603122i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.66 - 0.462i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.02 + 1.70i)T \) |
good | 2 | \( 1 - 2.32T + 2T^{2} \) |
| 11 | \( 1 + (-1.47 - 2.56i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.00 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.98 - 3.43i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.55 + 4.43i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.216 - 0.375i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.68 + 2.91i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 9.06T + 31T^{2} \) |
| 37 | \( 1 + (-0.0400 - 0.0693i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.435 + 0.753i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.02 - 1.77i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.84T + 47T^{2} \) |
| 53 | \( 1 + (4.24 - 7.35i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 9.09T + 67T^{2} \) |
| 71 | \( 1 + 2.95T + 71T^{2} \) |
| 73 | \( 1 + (5.84 - 10.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + (0.126 - 0.218i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.58 - 14.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.67 + 13.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.64694178228502322112896757028, −11.22351983789501804785464761289, −10.22114587285504352096393722925, −8.904056009609693654706516526862, −7.20930050050420936282222417950, −6.46590554189728977812006269432, −5.41834876737923611514040949699, −4.48498991556591572916130602674, −4.01324298016004244649750381645, −1.82724369097947077447647887003,
2.00164757017377973945560341029, 3.54226760939269210545549995296, 4.82741410929448607275564418514, 5.63990689491300386115542383386, 6.28656160986397473918479875960, 7.32238665337980444549845330393, 8.717318132912460806546945128015, 10.35487787187361387623262204373, 11.33610134157708554429568130860, 11.63312513707605515283809128222