L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 0.999·8-s − 11-s + 0.999·12-s + (−0.5 − 0.866i)16-s + (1 + 1.73i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)24-s − 27-s + (0.499 − 0.866i)32-s + (0.5 + 0.866i)33-s + (−0.999 + 1.73i)34-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 0.999·8-s − 11-s + 0.999·12-s + (−0.5 − 0.866i)16-s + (1 + 1.73i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)24-s − 27-s + (0.499 − 0.866i)32-s + (0.5 + 0.866i)33-s + (−0.999 + 1.73i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.005381333\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.005381333\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−8.455486710762704136513495198753, −7.908190608243251120909372638082, −7.46696163752520357293982429056, −6.53633365398466010734190753223, −5.91220156149461080410026856878, −5.58269055062843969766651745156, −4.40672017324912073846013203006, −3.69640326728159162261851970047, −2.61457635491670598605040696977, −1.30942933133815676285840403259,
0.54689964200041478865156603202, 2.20386241958171626950647796076, 2.92764953168571422288492782366, 3.91585151042771748337942254484, 4.67232190865074126792521798506, 5.32763178529595750794184896519, 5.66467494285632800173711656226, 6.94944209964752139845463856012, 7.67910070770663830479328246272, 8.801061667146172223900006302073