| L(s) = 1 | + (0.774 + 0.320i)2-s + (0.00133 + 1.73i)3-s + (−0.917 − 0.917i)4-s + (−0.159 − 0.802i)5-s + (−0.554 + 1.34i)6-s + (0.191 + 0.0380i)7-s + (−1.05 − 2.55i)8-s + (−2.99 + 0.00463i)9-s + (0.133 − 0.672i)10-s + (3.67 − 2.45i)11-s + (1.58 − 1.58i)12-s + (−3.75 + 3.75i)13-s + (0.136 + 0.0909i)14-s + (1.38 − 0.277i)15-s + 0.276i·16-s + (−2.89 + 2.93i)17-s + ⋯ |
| L(s) = 1 | + (0.547 + 0.226i)2-s + (0.000772 + 0.999i)3-s + (−0.458 − 0.458i)4-s + (−0.0713 − 0.358i)5-s + (−0.226 + 0.547i)6-s + (0.0723 + 0.0143i)7-s + (−0.374 − 0.902i)8-s + (−0.999 + 0.00154i)9-s + (0.0423 − 0.212i)10-s + (1.10 − 0.739i)11-s + (0.458 − 0.458i)12-s + (−1.04 + 1.04i)13-s + (0.0363 + 0.0243i)14-s + (0.358 − 0.0716i)15-s + 0.0690i·16-s + (−0.701 + 0.712i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.908748 + 0.258161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.908748 + 0.258161i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.00133 - 1.73i)T \) |
| 17 | \( 1 + (2.89 - 2.93i)T \) |
| good | 2 | \( 1 + (-0.774 - 0.320i)T + (1.41 + 1.41i)T^{2} \) |
| 5 | \( 1 + (0.159 + 0.802i)T + (-4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-0.191 - 0.0380i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-3.67 + 2.45i)T + (4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (3.75 - 3.75i)T - 13iT^{2} \) |
| 19 | \( 1 + (-0.210 + 0.508i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.29 - 3.43i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.606 + 0.120i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-0.369 + 0.553i)T + (-11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-1.63 - 1.09i)T + (14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-1.55 + 7.84i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (2.95 + 7.13i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-7.05 - 7.05i)T + 47iT^{2} \) |
| 53 | \( 1 + (10.7 + 4.43i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (0.0995 + 0.240i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (2.05 - 10.3i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + 5.40iT - 67T^{2} \) |
| 71 | \( 1 + (-3.61 + 5.41i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-11.3 + 2.26i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-6.00 - 8.98i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-1.05 + 2.55i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (3.69 - 3.69i)T - 89iT^{2} \) |
| 97 | \( 1 + (-0.481 - 2.42i)T + (-89.6 + 37.1i)T^{2} \) |
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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
−15.42939020469832035284056630619, −14.52062418228290079978889926711, −13.78037987273606237486041919731, −12.22766717203138001850021520930, −10.95388392035019329883961547634, −9.512733236206612700202399056418, −8.826416197463637415368115377997, −6.45325355165928309499655921892, −5.00773160805157211809756982919, −3.93511030605351382362489911689,
2.83980881454001982835488196894, 4.86442545256790819917966962500, 6.72599913041588450115626916559, 7.921370334924943884725257659689, 9.317362172313225985405825069597, 11.24650325261740251578003965582, 12.31273827084360222774381436698, 12.96294925731090844698797950468, 14.23341696426991141032208899401, 14.88558527547928096601797895698
