| L(s) = 1 | − 2i·3-s + (−1 + 2i)5-s − 2i·7-s − 9-s − 4·11-s + (4 + 2i)15-s + 8i·17-s − 19-s − 4·21-s + 6i·23-s + (−3 − 4i)25-s − 4i·27-s − 2·29-s + 8·31-s + 8i·33-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (−0.447 + 0.894i)5-s − 0.755i·7-s − 0.333·9-s − 1.20·11-s + (1.03 + 0.516i)15-s + 1.94i·17-s − 0.229·19-s − 0.872·21-s + 1.25i·23-s + (−0.600 − 0.800i)25-s − 0.769i·27-s − 0.371·29-s + 1.43·31-s + 1.39i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8770857671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8770857671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 8iT - 17T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−9.880782130745058808654517468042, −8.313884147095107406321305557223, −7.891301947340583955684355685581, −7.29263476837926517814888242162, −6.50275635170095270811662087602, −5.84351947352599318538626794631, −4.44448359996504954205454824091, −3.50437497519300771871528427078, −2.44349689500075743121062098094, −1.31385920126656720218383332207,
0.35771347266546139396829529225, 2.35559704477697595954339574485, 3.33865801319378317892570592342, 4.57385193052375166052643085086, 4.93260382947236744360248063708, 5.65265511249086114796470726652, 7.00092467746203929968661592773, 7.930612704967571769842427199455, 8.778707145911594731614301289917, 9.200774305934452121600986859692
