| L(s) = 1 | + 1.54i·2-s + i·3-s − 0.393·4-s + i·5-s − 1.54·6-s + (−2.51 + 0.814i)7-s + 2.48i·8-s − 9-s − 1.54·10-s + (1.71 − 2.83i)11-s − 0.393i·12-s − 2.83·13-s + (−1.26 − 3.89i)14-s − 15-s − 4.63·16-s − 1.11·17-s + ⋯ |
| L(s) = 1 | + 1.09i·2-s + 0.577i·3-s − 0.196·4-s + 0.447i·5-s − 0.631·6-s + (−0.951 + 0.308i)7-s + 0.878i·8-s − 0.333·9-s − 0.489·10-s + (0.518 − 0.855i)11-s − 0.113i·12-s − 0.785·13-s + (−0.336 − 1.04i)14-s − 0.258·15-s − 1.15·16-s − 0.269·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6753204302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6753204302\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (2.51 - 0.814i)T \) |
| 11 | \( 1 + (-1.71 + 2.83i)T \) |
| good | 2 | \( 1 - 1.54iT - 2T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 + 2.50iT - 29T^{2} \) |
| 31 | \( 1 - 4.63iT - 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 + 4.46T + 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 - 3.19iT - 47T^{2} \) |
| 53 | \( 1 + 3.06T + 53T^{2} \) |
| 59 | \( 1 + 4.55iT - 59T^{2} \) |
| 61 | \( 1 + 2.53T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 + 4.86T + 73T^{2} \) |
| 79 | \( 1 - 4.77iT - 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 0.588iT - 89T^{2} \) |
| 97 | \( 1 - 7.73iT - 97T^{2} \) |
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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
−10.19827999435689021541851173715, −9.459179030500166344002590397614, −8.630943813576695761457627947749, −7.888773822478010377812273916975, −6.81439828747788758482120299529, −6.29365535779169847713073994707, −5.60780298898777807974398381321, −4.50355898587636506783555159957, −3.35180706699102062614368354982, −2.38090425412065021123722780692,
0.26742119658745292865703083649, 1.71230944972389825545084400061, 2.55115514221623975043927264343, 3.72922647078823246789547847541, 4.52474504052600353504974864382, 5.94294987189690100941090372602, 6.82921198261741507381026456568, 7.37164459163911835486219235621, 8.595863441850875828952722974018, 9.576218651007528276161076389173
