L(s) = 1 | + 2i·2-s − 7.57i·3-s − 4·4-s + 15.1·6-s + 35.4i·7-s − 8i·8-s − 30.3·9-s − 16.6·11-s + 30.2i·12-s − 79.9i·13-s − 70.8·14-s + 16·16-s + 46.8i·17-s − 60.6i·18-s + 110.·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.45i·3-s − 0.5·4-s + 1.03·6-s + 1.91i·7-s − 0.353i·8-s − 1.12·9-s − 0.455·11-s + 0.728i·12-s − 1.70i·13-s − 1.35·14-s + 0.250·16-s + 0.667i·17-s − 0.794i·18-s + 1.33·19-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
\(L(2)\) |
\(\approx\) |
\(0.9260279340\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9260279340\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 7.57iT - 27T^{2} \) |
| 7 | \( 1 - 35.4iT - 343T^{2} \) |
| 11 | \( 1 + 16.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 46.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 110.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 0.836T + 2.43e4T^{2} \) |
| 31 | \( 1 + 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 368. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 95.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 331. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 535. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 409. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 352.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 507.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 820. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 733.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 91.4iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 329.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 753. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 271. iT - 9.12e5T^{2} \) |
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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
−9.306571885096082012437473936958, −8.617281340469096438096066671429, −7.78707632598614402622744290096, −7.48703399183750746617326331212, −6.05348892924179661203965477898, −5.87739900138694887134677382959, −5.05146444221839506856218980300, −3.15971390247806440884629944594, −2.38004787532276485882699157876, −1.11306784095856652707275930997,
0.24437804750581416126602011241, 1.59776118079782908265763964897, 3.25604697081130000963720479443, 3.83502215060235030318799679925, 4.61562536715902462447582439170, 5.18961192725800259287628224432, 6.80472610124201597058869453172, 7.49129473953787681323008679730, 8.707094334547733989042182393891, 9.569139690985744304188041992709