| L(s) = 1 | + 2.53i·2-s + i·3-s − 4.43·4-s − 2.53·6-s + 1.04i·7-s − 6.19i·8-s − 9-s + 2.97·11-s − 4.43i·12-s − 5.66i·13-s − 2.64·14-s + 6.83·16-s − 5.08i·17-s − 2.53i·18-s + 5.37·19-s + ⋯ |
| L(s) = 1 | + 1.79i·2-s + 0.577i·3-s − 2.21·4-s − 1.03·6-s + 0.393i·7-s − 2.18i·8-s − 0.333·9-s + 0.895·11-s − 1.28i·12-s − 1.57i·13-s − 0.705·14-s + 1.70·16-s − 1.23i·17-s − 0.598i·18-s + 1.23·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.366436912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366436912\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 2.53iT - 2T^{2} \) |
| 7 | \( 1 - 1.04iT - 7T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 + 5.66iT - 13T^{2} \) |
| 17 | \( 1 + 5.08iT - 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 + 3.86iT - 23T^{2} \) |
| 29 | \( 1 + 0.679T + 29T^{2} \) |
| 31 | \( 1 - 0.850T + 31T^{2} \) |
| 37 | \( 1 + 1.61iT - 37T^{2} \) |
| 41 | \( 1 - 1.16T + 41T^{2} \) |
| 43 | \( 1 + 5.68iT - 43T^{2} \) |
| 47 | \( 1 - 3.28iT - 47T^{2} \) |
| 53 | \( 1 + 12.6iT - 53T^{2} \) |
| 59 | \( 1 + 3.21T + 59T^{2} \) |
| 61 | \( 1 + 5.42T + 61T^{2} \) |
| 67 | \( 1 + 0.929iT - 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 + 11.3iT - 73T^{2} \) |
| 79 | \( 1 - 1.44T + 79T^{2} \) |
| 83 | \( 1 - 11.4iT - 83T^{2} \) |
| 89 | \( 1 + 9.07T + 89T^{2} \) |
| 97 | \( 1 + 6.02iT - 97T^{2} \) |
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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
−9.249781503929315256522302489120, −8.508337290933394314634702902734, −7.77545477120234156461915502648, −7.07643745902700609612302686346, −6.21181597646341764364863531204, −5.41000899711651088858648081152, −4.99231151932615563195233414445, −3.91748862617649803265502134519, −2.90414363961249670623488603184, −0.59242077661440399393963836408,
1.22796670126649240536462245496, 1.71070072929593261902882116411, 2.93869466149655848502294024055, 3.89421045776386897536304398106, 4.41879553237391627365874265945, 5.68382741413752426454997723746, 6.68936591116922847232611493900, 7.58179623870455721356965958219, 8.637152168984508140328931093341, 9.271731225616935042554552957679
