| L(s) = 1 | − 1.79i·2-s + i·3-s − 1.20·4-s + 1.79·6-s − i·7-s − 1.41i·8-s − 9-s + 5·11-s − 1.20i·12-s − 4.58i·13-s − 1.79·14-s − 4.95·16-s + 3i·17-s + 1.79i·18-s − 3.58·19-s + ⋯ |
| L(s) = 1 | − 1.26i·2-s + 0.577i·3-s − 0.604·4-s + 0.731·6-s − 0.377i·7-s − 0.501i·8-s − 0.333·9-s + 1.50·11-s − 0.348i·12-s − 1.27i·13-s − 0.478·14-s − 1.23·16-s + 0.727i·17-s + 0.422i·18-s − 0.821·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.658103881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658103881\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.79iT - 2T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 4.58iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 9.16T + 41T^{2} \) |
| 43 | \( 1 + 9.58iT - 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 0.417iT - 53T^{2} \) |
| 59 | \( 1 - 7.58T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 4.16iT - 67T^{2} \) |
| 71 | \( 1 + 9.58T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 7.58T + 79T^{2} \) |
| 83 | \( 1 + 11.5iT - 83T^{2} \) |
| 89 | \( 1 + 1.41T + 89T^{2} \) |
| 97 | \( 1 + 11.5iT - 97T^{2} \) |
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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
−8.753101601478421218480050782184, −8.476225949962167607138452332631, −7.04388804678459295085891399283, −6.42693224397819184236512680534, −5.35506857290199480795398613436, −4.16357382058152508737311638387, −3.82146709273995111677361717207, −2.87039569639614589938371165287, −1.76332515639060866721098504392, −0.58698656628805986390457317485,
1.46564327400613206822686900070, 2.49355935340586187300617968016, 3.92994855302328972088214128811, 4.79646561414338014838608766048, 5.78772599731370243859779649171, 6.52025793371046685441787730877, 6.85675038928827682182166198132, 7.65522005202914575443512844989, 8.552990298507929512903568142319, 9.065106364759053554227844637744
