| 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
−10.90541719282905949210954696085, −10.19214229021819033751593806933, −9.219892882875995952715669714356, −7.65301102371440735832546969990, −6.87927545992574621630155473055, −6.59073746030475107884448841587, −5.31607747052952208049449339600, −3.63201327232890339777801597758, −2.38315794005604406609735775884, −0.46870070659717412470538283649,
2.18221931742084801453043007983, 4.05839526650636296276696324050, 4.67032228202979270846491131032, 5.94735901967266165280349610838, 6.45485641570394548334253100419, 8.423617801241566768641030656596, 9.053231476647913982653393858975, 9.780707047586195411481710773248, 10.58435583558097436215807865757, 11.69414996358297948696745938677
