L(s) = 1 | − i·2-s − i·3-s + 4-s − 6-s + 2i·7-s − 3i·8-s − 9-s − 2·11-s − i·12-s − 4i·13-s + 2·14-s − 16-s − 2i·17-s + i·18-s + 19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s + 0.5·4-s − 0.408·6-s + 0.755i·7-s − 1.06i·8-s − 0.333·9-s − 0.603·11-s − 0.288i·12-s − 1.10i·13-s + 0.534·14-s − 0.250·16-s − 0.485i·17-s + 0.235i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.611209360\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.611209360\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−9.245530786124772865306343605741, −8.396825128819813699431041319497, −7.54860048137476227858027593375, −6.84651831548842732552859143365, −5.83823170846390226942797909339, −5.18833476572896479929421032048, −3.65889501089787936484376911937, −2.72159519706506970294224739056, −2.08213217570442925331548106914, −0.61126735881693447820401088731,
1.67676595232616429858009348770, 2.96057790387416422175793007869, 4.06379458036478528891979621343, 4.94268950445029901429194254953, 5.87680264168253826903866770543, 6.61875279334822798305403619151, 7.52516610574022211698238937970, 7.999753345338607201275445207732, 9.109881695958290361830136126141, 9.783362414971612688989088021925