| L(s) = 1 | − 0.539·3-s − 0.460i·5-s + i·7-s − 2.70·9-s − 0.829i·11-s + (−2.87 − 2.17i)13-s + 0.248i·15-s + 2.87·17-s + 4.32i·19-s − 0.539i·21-s + 5.04·23-s + 4.78·25-s + 3.07·27-s + 0.261·29-s + 6.80i·31-s + ⋯ |
| L(s) = 1 | − 0.311·3-s − 0.206i·5-s + 0.377i·7-s − 0.903·9-s − 0.250i·11-s + (−0.798 − 0.601i)13-s + 0.0641i·15-s + 0.698·17-s + 0.992i·19-s − 0.117i·21-s + 1.05·23-s + 0.957·25-s + 0.592·27-s + 0.0486·29-s + 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.175442543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.175442543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (2.87 + 2.17i)T \) |
| good | 3 | \( 1 + 0.539T + 3T^{2} \) |
| 5 | \( 1 + 0.460iT - 5T^{2} \) |
| 11 | \( 1 + 0.829iT - 11T^{2} \) |
| 17 | \( 1 - 2.87T + 17T^{2} \) |
| 19 | \( 1 - 4.32iT - 19T^{2} \) |
| 23 | \( 1 - 5.04T + 23T^{2} \) |
| 29 | \( 1 - 0.261T + 29T^{2} \) |
| 31 | \( 1 - 6.80iT - 31T^{2} \) |
| 37 | \( 1 - 9.51iT - 37T^{2} \) |
| 41 | \( 1 - 6.68iT - 41T^{2} \) |
| 43 | \( 1 + 0.418T + 43T^{2} \) |
| 47 | \( 1 + 9.24iT - 47T^{2} \) |
| 53 | \( 1 - 1.63T + 53T^{2} \) |
| 59 | \( 1 - 2.78iT - 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 + 5.07iT - 67T^{2} \) |
| 71 | \( 1 - 12.7iT - 71T^{2} \) |
| 73 | \( 1 - 0.353iT - 73T^{2} \) |
| 79 | \( 1 + 2.81T + 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 5.43iT - 89T^{2} \) |
| 97 | \( 1 - 12.6iT - 97T^{2} \) |
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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
−9.727407001962422913533691092048, −8.613492552436201041848428585849, −8.280541896053846024512744699301, −7.17054716964200965025792064719, −6.29216175045846442873491656423, −5.33846463915139323724964527824, −4.96043796384357736110118931897, −3.40837539028587995268273255430, −2.69453015020525254771308147651, −1.08824254876877395266273774873,
0.58029520829532333251668410503, 2.28946354071416587610606244139, 3.21513219141069085811710486734, 4.44849047817613504636305579098, 5.20926722409444685144594183349, 6.11687568792733096953552054529, 7.07749309900906762945671842586, 7.56352229810701786367878436974, 8.778030766020012938464049946070, 9.310773688398195143135225469581
