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
−9.783362414971612688989088021925, −9.109881695958290361830136126141, −7.999753345338607201275445207732, −7.52516610574022211698238937970, −6.61875279334822798305403619151, −5.87680264168253826903866770543, −4.94268950445029901429194254953, −4.06379458036478528891979621343, −2.96057790387416422175793007869, −1.67676595232616429858009348770,
0.61126735881693447820401088731, 2.08213217570442925331548106914, 2.72159519706506970294224739056, 3.65889501089787936484376911937, 5.18833476572896479929421032048, 5.83823170846390226942797909339, 6.84651831548842732552859143365, 7.54860048137476227858027593375, 8.396825128819813699431041319497, 9.245530786124772865306343605741