| L(s) = 1 | + (−0.962 + 1.88i)3-s + (0.712 − 2.11i)5-s + (−2.48 + 2.48i)7-s + (−0.876 − 1.20i)9-s + (0.636 − 0.876i)11-s + (−0.803 + 5.07i)13-s + (3.31 + 3.38i)15-s + (−5.83 + 2.97i)17-s + (−1.78 + 5.48i)19-s + (−2.30 − 7.08i)21-s + (−1.03 − 6.54i)23-s + (−3.98 − 3.02i)25-s + (−3.15 + 0.500i)27-s + (5.38 − 1.75i)29-s + (−1.28 − 0.416i)31-s + ⋯ |
| L(s) = 1 | + (−0.555 + 1.09i)3-s + (0.318 − 0.947i)5-s + (−0.939 + 0.939i)7-s + (−0.292 − 0.402i)9-s + (0.192 − 0.264i)11-s + (−0.222 + 1.40i)13-s + (0.856 + 0.874i)15-s + (−1.41 + 0.720i)17-s + (−0.409 + 1.25i)19-s + (−0.502 − 1.54i)21-s + (−0.216 − 1.36i)23-s + (−0.796 − 0.604i)25-s + (−0.607 + 0.0962i)27-s + (1.00 − 0.325i)29-s + (−0.230 − 0.0748i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.196573 + 0.676988i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.196573 + 0.676988i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.712 + 2.11i)T \) |
| good | 3 | \( 1 + (0.962 - 1.88i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (2.48 - 2.48i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.636 + 0.876i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.803 - 5.07i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (5.83 - 2.97i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (1.78 - 5.48i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.03 + 6.54i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-5.38 + 1.75i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.28 + 0.416i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.392 - 0.0620i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (5.94 - 4.31i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.79 - 5.79i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.36 - 0.697i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-5.76 - 2.93i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (4.21 - 3.06i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.55 - 6.21i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (0.158 + 0.310i)T + (-39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-6.11 + 1.98i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.41 + 0.540i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.562 - 1.73i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-9.67 + 4.93i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (1.98 - 2.73i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.35 + 4.61i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−11.69414996358297948696745938677, −10.58435583558097436215807865757, −9.780707047586195411481710773248, −9.053231476647913982653393858975, −8.423617801241566768641030656596, −6.45485641570394548334253100419, −5.94735901967266165280349610838, −4.67032228202979270846491131032, −4.05839526650636296276696324050, −2.18221931742084801453043007983,
0.46870070659717412470538283649, 2.38315794005604406609735775884, 3.63201327232890339777801597758, 5.31607747052952208049449339600, 6.59073746030475107884448841587, 6.87927545992574621630155473055, 7.65301102371440735832546969990, 9.219892882875995952715669714356, 10.19214229021819033751593806933, 10.90541719282905949210954696085
