L(s) = 1 | + (0.448 − 1.67i)3-s + (−1.73 + i)4-s + (0.189 + 2.63i)7-s + (−2.59 − 1.50i)9-s + (0.896 + 3.34i)12-s + (4.89 + 4.89i)13-s + (1.99 − 3.46i)16-s + (6.92 + 4i)19-s + (4.50 + 0.866i)21-s + (−3.67 + 3.67i)27-s + (−2.96 − 4.38i)28-s + (−3.5 − 6.06i)31-s + 6·36-s + (1.34 + 5.01i)37-s + (10.3 − 6i)39-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)3-s + (−0.866 + 0.5i)4-s + (0.0716 + 0.997i)7-s + (−0.866 − 0.5i)9-s + (0.258 + 0.965i)12-s + (1.35 + 1.35i)13-s + (0.499 − 0.866i)16-s + (1.58 + 0.917i)19-s + (0.981 + 0.188i)21-s + (−0.707 + 0.707i)27-s + (−0.560 − 0.827i)28-s + (−0.628 − 1.08i)31-s + 36-s + (0.221 + 0.825i)37-s + (1.66 − 0.960i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23831 + 0.294879i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23831 + 0.294879i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.448 + 1.67i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.189 - 2.63i)T \) |
good | 2 | \( 1 + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.89 - 4.89i)T + 13iT^{2} \) |
| 17 | \( 1 + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-6.92 - 4i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.34 - 5.01i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-8.57 - 8.57i)T + 43iT^{2} \) |
| 47 | \( 1 + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.34 - 0.896i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (0.448 - 1.67i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (14.7 + 8.5i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.67 + 3.67i)T - 97iT^{2} \) |
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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
−11.38420091019427001481609761423, −9.592972153111262260018271617719, −9.056286482656038004168503835863, −8.268195537121943742124075619364, −7.53812592438099509352609599492, −6.28836015723211368458602126801, −5.50487132403223815033590005044, −4.05567267908212512396076764298, −2.97219976236701242639700440210, −1.46472020393295575028365369147,
0.866404681991786976776496362435, 3.25177775853202690774510881919, 4.00691147074192089685643714493, 5.11825562724216682660506086223, 5.77391138178279621155975838917, 7.34668101451046793403080864168, 8.368308989940967166818592667592, 9.112439107446540772243682580315, 9.957131323603720704325811845198, 10.68597247580603178185822719699