| L(s) = 1 | − 6.72i·2-s − 9i·3-s − 13.2·4-s − 60.5·6-s + 49i·7-s − 126. i·8-s − 81·9-s + 255.·11-s + 119. i·12-s − 404. i·13-s + 329.·14-s − 1.27e3·16-s − 1.15e3i·17-s + 544. i·18-s + 1.87e3·19-s + ⋯ |
| L(s) = 1 | − 1.18i·2-s − 0.577i·3-s − 0.413·4-s − 0.686·6-s + 0.377i·7-s − 0.697i·8-s − 0.333·9-s + 0.636·11-s + 0.238i·12-s − 0.663i·13-s + 0.449·14-s − 1.24·16-s − 0.972i·17-s + 0.396i·18-s + 1.19·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.170309705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.170309705\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49iT \) |
| good | 2 | \( 1 + 6.72iT - 32T^{2} \) |
| 11 | \( 1 - 255.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 404. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.15e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.87e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 425. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.44e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 2.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.53e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.95e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.38e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.12e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.00e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.92e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 4.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.19e4iT - 8.58e9T^{2} \) |
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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
−9.673041594285836689951476930936, −8.964592874990559543322484071822, −7.75275560884892022595288778920, −6.87429196947250289566669390996, −5.85027131702324680416363386732, −4.61144444751683189153651672412, −3.25827025167749894964131722805, −2.58925446570479111881741485995, −1.35237066485135011035717462833, −0.52316293243293751890867984694,
1.32896116750793419546143829976, 2.95754503984303266987849391151, 4.26872830681149619141655136174, 5.04321289430172944140653513661, 6.26392427551215945572622456526, 6.72741176358194081279902691093, 7.920462146720034399219562053514, 8.555829003724235140726572244234, 9.577736411758450485306241370151, 10.39932448674539202909140917877
