| L(s) = 1 | + 1.23i·3-s + 0.236i·7-s + 1.47·9-s + 11-s − 2.23i·13-s − 2.47i·17-s − 19-s − 0.291·21-s + i·23-s + 5.52i·27-s + 6.23·29-s − 8.47·31-s + 1.23i·33-s + 6.76i·37-s + 2.76·39-s + ⋯ |
| L(s) = 1 | + 0.713i·3-s + 0.0892i·7-s + 0.490·9-s + 0.301·11-s − 0.620i·13-s − 0.599i·17-s − 0.229·19-s − 0.0636·21-s + 0.208i·23-s + 1.06i·27-s + 1.15·29-s − 1.52·31-s + 0.215i·33-s + 1.11i·37-s + 0.442·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.099200085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.099200085\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 - 1.23iT - 3T^{2} \) |
| 7 | \( 1 - 0.236iT - 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 2.23iT - 13T^{2} \) |
| 17 | \( 1 + 2.47iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 - 6.76iT - 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 11.4iT - 43T^{2} \) |
| 47 | \( 1 + 1.70iT - 47T^{2} \) |
| 53 | \( 1 - 3.23iT - 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 + 4.94iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 - 0.527iT - 73T^{2} \) |
| 79 | \( 1 - 7.18T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 16.1iT - 97T^{2} \) |
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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
−8.525822970655799763427615386917, −7.54720235534033804938995379990, −7.05364703761660505664492770827, −6.09650644538525204344091050331, −5.32588731330798962974952497215, −4.63355156205046568612357614609, −3.87829425719174037918239165735, −3.12389559337084299398196648694, −2.05923097985491123177103830134, −0.799973499718488600341583529512,
0.861547556275380727622899041899, 1.79175041630982347049259360433, 2.61261581539510535648244937039, 3.89867415332916577072895792677, 4.34887953236486668901650709436, 5.43618571507306981409759320994, 6.32019320138899204081364672720, 6.75647617338326093496122084601, 7.56964521892321194976502745978, 8.063375004310325211272214583965
