L(s) = 1 | − 1.41i·2-s − 2.00·4-s + (−0.158 + 2.23i)5-s + 2.82i·8-s + 3·9-s + (3.15 + 0.224i)10-s − 5.99·13-s + 4.00·16-s − 6.62·17-s − 4.24i·18-s + (0.317 − 4.46i)20-s + (−4.94 − 0.707i)25-s + 8.47i·26-s − 9.89·29-s − 5.65i·32-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s + (−0.0708 + 0.997i)5-s + 1.00i·8-s + 9-s + (0.997 + 0.0708i)10-s − 1.66·13-s + 1.00·16-s − 1.60·17-s − 0.999i·18-s + (0.0708 − 0.997i)20-s + (−0.989 − 0.141i)25-s + 1.66i·26-s − 1.83·29-s − 1.00i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.190934 + 0.273106i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.190934 + 0.273106i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 + (0.158 - 2.23i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5.99T + 13T^{2} \) |
| 17 | \( 1 + 6.62T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.07iT - 37T^{2} \) |
| 41 | \( 1 - 3.56iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 14iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 3.11iT - 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−10.34151914262841524371682797757, −9.622723908561502602009734541388, −8.963953257463201638917434552991, −7.54179259950133573970286747738, −7.17176065036849673018925521979, −5.86925510671880584393435267374, −4.64201159662751524417689683825, −3.97363962383103706354742740207, −2.70597825221033768923506814197, −1.93866635695927036827263840947,
0.14639041080762727431872217361, 1.91863374377822642920842364650, 3.86169173375896259579069010958, 4.71781622134257662350376452112, 5.20597830701360282994898312788, 6.48153321729390999038324150784, 7.25240856761865723176643275987, 7.931909517429951765379381641439, 8.919262112733134515747376661610, 9.517297205484019135802920379109