L(s) = 1 | + 2.54i·3-s + (−1.30 + 1.81i)5-s − 0.780i·7-s − 3.45·9-s − 5.16·11-s − 3.34i·13-s + (−4.60 − 3.32i)15-s − 6.62i·17-s + 6.40·19-s + 1.98·21-s + i·23-s + (−1.58 − 4.74i)25-s − 1.15i·27-s + 0.0877·29-s − 3.29·31-s + ⋯ |
L(s) = 1 | + 1.46i·3-s + (−0.584 + 0.811i)5-s − 0.295i·7-s − 1.15·9-s − 1.55·11-s − 0.928i·13-s + (−1.19 − 0.857i)15-s − 1.60i·17-s + 1.47·19-s + 0.432·21-s + 0.208i·23-s + (−0.316 − 0.948i)25-s − 0.223i·27-s + 0.0162·29-s − 0.591·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6562005768\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6562005768\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.30 - 1.81i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 2.54iT - 3T^{2} \) |
| 7 | \( 1 + 0.780iT - 7T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 13 | \( 1 + 3.34iT - 13T^{2} \) |
| 17 | \( 1 + 6.62iT - 17T^{2} \) |
| 19 | \( 1 - 6.40T + 19T^{2} \) |
| 29 | \( 1 - 0.0877T + 29T^{2} \) |
| 31 | \( 1 + 3.29T + 31T^{2} \) |
| 37 | \( 1 + 4.68iT - 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 - 2.62iT - 43T^{2} \) |
| 47 | \( 1 + 5.13iT - 47T^{2} \) |
| 53 | \( 1 + 8.80iT - 53T^{2} \) |
| 59 | \( 1 - 2.98T + 59T^{2} \) |
| 61 | \( 1 + 2.05T + 61T^{2} \) |
| 67 | \( 1 - 3.60iT - 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 3.43T + 79T^{2} \) |
| 83 | \( 1 - 3.62iT - 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 0.427iT - 97T^{2} \) |
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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
−9.454084402599926191836505322516, −8.388173102216514847586276289787, −7.54591036487371624760417260622, −7.08084365951907157107446849377, −5.37864374189263920865448991016, −5.29822675003483342374437401741, −4.12953140994958946782316726123, −3.20441019827073668030850251636, −2.75593404371519361741063099427, −0.25895704574645096630713320565,
1.20158731750437770022328276126, 2.08916649089591664265629657503, 3.26092148764927780399940606308, 4.49753528919587766114125429316, 5.44681861879970225607919520006, 6.14090776112958474331938150184, 7.23116933191783344644045338334, 7.69935482487405679750679436505, 8.360381912923051296223038806612, 8.994052837031760512937492468226