| L(s) = 1 | + i·3-s − 2.44i·7-s − 9-s − 0.449i·13-s + 2i·17-s − 4.89·19-s + 2.44·21-s − 6.89i·23-s − i·27-s + 4.44·29-s + 31-s − 8.44i·37-s + 0.449·39-s − 11.7·41-s + 12.8i·43-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.925i·7-s − 0.333·9-s − 0.124i·13-s + 0.485i·17-s − 1.12·19-s + 0.534·21-s − 1.43i·23-s − 0.192i·27-s + 0.826·29-s + 0.179·31-s − 1.38i·37-s + 0.0719·39-s − 1.84·41-s + 1.96i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3911815124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3911815124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.449iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 + 6.89iT - 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 37 | \( 1 + 8.44iT - 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 12.8iT - 43T^{2} \) |
| 47 | \( 1 + 0.898iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 - 2.89T + 61T^{2} \) |
| 67 | \( 1 + 1.55iT - 67T^{2} \) |
| 71 | \( 1 - 6.44T + 71T^{2} \) |
| 73 | \( 1 - 7.55iT - 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 + 0.449T + 89T^{2} \) |
| 97 | \( 1 + 14.8iT - 97T^{2} \) |
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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
−8.199044479109129431182020431909, −7.28245521327212268787004072458, −6.55843279037680082928850775056, −6.08838881944304988703660310903, −5.07457419250107867299805681852, −4.40055078120175865314570193057, −3.97707844917958713769105924308, −3.05798655101282022371662981803, −2.21255299633185449729806515224, −1.03500861272833061898140514337,
0.092723193434503146768041818815, 1.46741105569673774391405823946, 2.17832604468212772676523797417, 2.98948706098254082014379108805, 3.76587346667737063737597610900, 4.86166721912674569871549596122, 5.34568950735058480235066511636, 6.17100812032299977230135369231, 6.69611370315050052950778234623, 7.36692032426340924645577609821
