| L(s) = 1 | − 3-s + 2.46i·5-s + i·7-s + 9-s + 0.836i·11-s + (3.51 − 0.812i)13-s − 2.46i·15-s + 2.16·17-s + 6.24i·19-s − i·21-s − 0.863·23-s − 1.05·25-s − 27-s + 4.16·29-s − 0.836i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.10i·5-s + 0.377i·7-s + 0.333·9-s + 0.252i·11-s + (0.974 − 0.225i)13-s − 0.635i·15-s + 0.524·17-s + 1.43i·19-s − 0.218i·21-s − 0.180·23-s − 0.211·25-s − 0.192·27-s + 0.772·29-s − 0.145i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.627818540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.627818540\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-3.51 + 0.812i)T \) |
| good | 5 | \( 1 - 2.46iT - 5T^{2} \) |
| 11 | \( 1 - 0.836iT - 11T^{2} \) |
| 17 | \( 1 - 2.16T + 17T^{2} \) |
| 19 | \( 1 - 6.24iT - 19T^{2} \) |
| 23 | \( 1 + 0.863T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 1.23iT - 37T^{2} \) |
| 41 | \( 1 - 6.95iT - 41T^{2} \) |
| 43 | \( 1 - 5.32T + 43T^{2} \) |
| 47 | \( 1 - 4.99iT - 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 + 10.0iT - 59T^{2} \) |
| 61 | \( 1 - 9.87T + 61T^{2} \) |
| 67 | \( 1 + 9.92iT - 67T^{2} \) |
| 71 | \( 1 + 3.68iT - 71T^{2} \) |
| 73 | \( 1 + 8.62iT - 73T^{2} \) |
| 79 | \( 1 - 4.30T + 79T^{2} \) |
| 83 | \( 1 - 6.60iT - 83T^{2} \) |
| 89 | \( 1 - 1.10iT - 89T^{2} \) |
| 97 | \( 1 - 13.9iT - 97T^{2} \) |
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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
−8.360953814241005317137684151261, −7.86850437994291330324206049496, −6.98915911220297610124023739640, −6.27405977512416469645999302878, −5.86895541812060718282351412609, −4.97408859567803881511390492537, −3.90158177960975590288247364357, −3.26037518439179903904391616473, −2.25533131418511598444333136094, −1.12660893524877218311788808261,
0.62134174653840066351478446386, 1.25299145675255692700382357173, 2.60199836646155539192781946001, 3.83799564141351308302061862083, 4.41324068800430640436362563500, 5.27753570101293974768495760483, 5.77072370528630740175334247888, 6.75925510552480371626265944893, 7.28702843046366810734953015734, 8.440728546253428766289678047179
