L(s) = 1 | − 2i·2-s + 4.74i·3-s − 4·4-s + 9.49·6-s + 29.3i·7-s + 8i·8-s + 4.44·9-s − 38.1·11-s − 18.9i·12-s + 22.5i·13-s + 58.7·14-s + 16·16-s − 104. i·17-s − 8.89i·18-s − 141.·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.913i·3-s − 0.5·4-s + 0.646·6-s + 1.58i·7-s + 0.353i·8-s + 0.164·9-s − 1.04·11-s − 0.456i·12-s + 0.480i·13-s + 1.12·14-s + 0.250·16-s − 1.48i·17-s − 0.116i·18-s − 1.71·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2435851178\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2435851178\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23iT \) |
good | 3 | \( 1 - 4.74iT - 27T^{2} \) |
| 7 | \( 1 - 29.3iT - 343T^{2} \) |
| 11 | \( 1 + 38.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 104. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 141.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 99.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 59.9iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 163. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 205. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 491. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 433.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 660.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 323. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 893.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 196. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 500.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 800. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 729.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3iT - 9.12e5T^{2} \) |
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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
−9.349466699447096663107953247402, −8.702364419353411016194709307799, −7.76789336624299573276810649644, −6.47563205221732268201684172268, −5.32915218728611544938779293053, −4.88376055798494515836811788501, −3.82894611157484328391020014665, −2.69216156754354786340274934855, −2.02321700426243220615038303503, −0.06567471794071414615777394119,
0.978073184713639528164462533027, 2.18753232541448796712264577026, 3.82230631453057098115959346987, 4.43686054349442100038109914231, 5.74890293314414450475458580775, 6.51550640811371884531618863074, 7.26973166335987193774360735405, 7.88145733873188362345653012695, 8.390868392663685600426997662253, 9.730882908076028824722682872370