| L(s) = 1 | − i·2-s − 4-s − 1.26·5-s + (1.09 − 2.40i)7-s + i·8-s + 1.26i·10-s + 2.94i·11-s + 0.580i·13-s + (−2.40 − 1.09i)14-s + 16-s + 0.124·17-s − i·19-s + 1.26·20-s + 2.94·22-s + 5.45i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.564·5-s + (0.413 − 0.910i)7-s + 0.353i·8-s + 0.399i·10-s + 0.888i·11-s + 0.161i·13-s + (−0.643 − 0.292i)14-s + 0.250·16-s + 0.0303·17-s − 0.229i·19-s + 0.282·20-s + 0.628·22-s + 1.13i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.220331310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.220331310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.09 + 2.40i)T \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.26T + 5T^{2} \) |
| 11 | \( 1 - 2.94iT - 11T^{2} \) |
| 13 | \( 1 - 0.580iT - 13T^{2} \) |
| 17 | \( 1 - 0.124T + 17T^{2} \) |
| 23 | \( 1 - 5.45iT - 23T^{2} \) |
| 29 | \( 1 - 3.23iT - 29T^{2} \) |
| 31 | \( 1 - 2.94iT - 31T^{2} \) |
| 37 | \( 1 + 0.135T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 6.95T + 47T^{2} \) |
| 53 | \( 1 - 5.46iT - 53T^{2} \) |
| 59 | \( 1 + 6.29T + 59T^{2} \) |
| 61 | \( 1 + 0.467iT - 61T^{2} \) |
| 67 | \( 1 - 5.93T + 67T^{2} \) |
| 71 | \( 1 - 16.1iT - 71T^{2} \) |
| 73 | \( 1 - 11.4iT - 73T^{2} \) |
| 79 | \( 1 - 0.468T + 79T^{2} \) |
| 83 | \( 1 - 9.79T + 83T^{2} \) |
| 89 | \( 1 - 9.24T + 89T^{2} \) |
| 97 | \( 1 + 7.54iT - 97T^{2} \) |
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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
−9.134978450936486207991108199057, −8.195632994759939479692382840780, −7.47036685772305317902860280046, −6.97265430769155799699895865981, −5.67290251092079174252469193173, −4.72790273643582310437640080382, −4.10306897077807148439388878791, −3.34576656646382227543814121649, −2.08715165482859705473343368051, −1.06214024148346648814362848033,
0.49074705908051948410177330698, 2.16755348697814887597192807537, 3.30230866532762017842428982896, 4.24766573155362407212495564675, 5.09790319055743067277116718612, 5.95452639755416778990865109992, 6.43030822249249214156537652357, 7.71463660650244081778586072656, 7.999769576049366392436991257830, 8.808326163245120547451055151003
