L(s) = 1 | − 8.27i·2-s − 25.6i·3-s − 36.4·4-s − 212.·6-s − 49i·7-s + 37.0i·8-s − 414.·9-s − 270.·11-s + 935. i·12-s + 300. i·13-s − 405.·14-s − 860.·16-s − 613. i·17-s + 3.43e3i·18-s + 1.70e3·19-s + ⋯ |
L(s) = 1 | − 1.46i·2-s − 1.64i·3-s − 1.13·4-s − 2.40·6-s − 0.377i·7-s + 0.204i·8-s − 1.70·9-s − 0.673·11-s + 1.87i·12-s + 0.493i·13-s − 0.552·14-s − 0.840·16-s − 0.514i·17-s + 2.49i·18-s + 1.08·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8071605880\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8071605880\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + 49iT \) |
good | 2 | \( 1 + 8.27iT - 32T^{2} \) |
| 3 | \( 1 + 25.6iT - 243T^{2} \) |
| 11 | \( 1 + 270.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 300. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 613. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.18e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 4.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.15e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 7.14e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.95e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.99e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.94e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.05e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.11e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 8.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.18e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 4.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.36e4iT - 8.58e9T^{2} \) |
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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
−11.33231484497451138146351427740, −10.11918089434484023205093207498, −9.017335446730545401254791259322, −7.63834038668581608499324708238, −6.93467580918626380236630044543, −5.38155509493841973191704957488, −3.55482997455285191228931430558, −2.33162275356981750289669815764, −1.37031054122088129466497359234, −0.26095649717168051161530436504,
2.98495650882085153740970856196, 4.50489662729762552978702557329, 5.29522454686024614241209559123, 6.18774542718380142226486971078, 7.73358857736965973562786077151, 8.600291330203699525720279327309, 9.548767521558910495323003118401, 10.48210781141559975756454126206, 11.48855466034233747024999449531, 13.01850915445719034752691842037