L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + 5.12i·7-s + i·8-s − 9-s + 5.12·11-s + i·12-s + 2i·13-s + 5.12·14-s + 16-s − 7.12i·17-s + i·18-s − 4·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.93i·7-s + 0.353i·8-s − 0.333·9-s + 1.54·11-s + 0.288i·12-s + 0.554i·13-s + 1.36·14-s + 0.250·16-s − 1.72i·17-s + 0.235i·18-s − 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.507645832\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.507645832\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 - 5.12iT - 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 7.12iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.12iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 - 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 + 3.12T + 89T^{2} \) |
| 97 | \( 1 + 0.246iT - 97T^{2} \) |
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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
−8.722457995193821617594679643176, −8.251046633550159255519468716978, −6.99147557093146811223111561719, −6.43783773808062316460392684465, −5.60136926833259158165553973156, −4.86170573229581804429762824791, −3.84750622133712027756308148593, −2.74730712632429857836956326375, −2.20664352222468011985422042775, −1.19027299419765442153715000695,
0.50093876815806335363385456539, 1.68485164808227681989585066417, 3.58033405128142892823340861114, 3.95593281602730264158125172457, 4.46056600830069372358049403947, 5.63063195866398608836386577653, 6.45832428338448985155961531899, 6.91681530437523969318487135242, 7.81636017929584628300719789821, 8.426751138366929919528912567523