L(s) = 1 | − 3-s − 3.33i·5-s + (1.56 − 2.13i)7-s + 9-s + 0.936i·11-s − 1.87i·13-s + 3.33i·15-s − 5.20i·17-s + 7.12·19-s + (−1.56 + 2.13i)21-s + 0.936i·23-s − 6.12·25-s − 27-s + 2·29-s − 0.936i·33-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.49i·5-s + (0.590 − 0.807i)7-s + 0.333·9-s + 0.282i·11-s − 0.519i·13-s + 0.861i·15-s − 1.26i·17-s + 1.63·19-s + (−0.340 + 0.466i)21-s + 0.195i·23-s − 1.22·25-s − 0.192·27-s + 0.371·29-s − 0.163i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.339414164\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.339414164\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + (-1.56 + 2.13i)T \) |
good | 5 | \( 1 + 3.33iT - 5T^{2} \) |
| 11 | \( 1 - 0.936iT - 11T^{2} \) |
| 13 | \( 1 + 1.87iT - 13T^{2} \) |
| 17 | \( 1 + 5.20iT - 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 - 0.936iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 + 1.46iT - 41T^{2} \) |
| 43 | \( 1 - 9.06iT - 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 4.79iT - 61T^{2} \) |
| 67 | \( 1 + 10.9iT - 67T^{2} \) |
| 71 | \( 1 - 3.86iT - 71T^{2} \) |
| 73 | \( 1 - 6.67iT - 73T^{2} \) |
| 79 | \( 1 - 2.39iT - 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 1.46iT - 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−9.581163708963897463510215831939, −8.431768940799101881743593974664, −7.72206259570269768405052442017, −7.02403455940168139442476216979, −5.74028782784380603773252728103, −4.90390205939182227464557877729, −4.63884101097916424146149123477, −3.27186977650580084886127669876, −1.48349689592514403211363225787, −0.64854587076054971189441186493,
1.65091388823882080071309704809, 2.81058993810904649015915106080, 3.75367748425564844978026577241, 5.00341239518922836017698192603, 5.87972296949063765393782322068, 6.52150724173690645471708573662, 7.35475742405766762712240911349, 8.170651919418833045734745493954, 9.153423385441413610885078119427, 10.11369794403117692196035203398