L(s) = 1 | − 2-s + 4-s − 5-s + 0.376i·7-s − 8-s + 10-s + 3.43·11-s − 3.04i·13-s − 0.376i·14-s + 16-s + 5.53i·17-s + 4.42·19-s − 20-s − 3.43·22-s + 2.14i·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.142i·7-s − 0.353·8-s + 0.316·10-s + 1.03·11-s − 0.843i·13-s − 0.100i·14-s + 0.250·16-s + 1.34i·17-s + 1.01·19-s − 0.223·20-s − 0.732·22-s + 0.446i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.340918755\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.340918755\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (7.52 + 3.21i)T \) |
good | 7 | \( 1 - 0.376iT - 7T^{2} \) |
| 11 | \( 1 - 3.43T + 11T^{2} \) |
| 13 | \( 1 + 3.04iT - 13T^{2} \) |
| 17 | \( 1 - 5.53iT - 17T^{2} \) |
| 19 | \( 1 - 4.42T + 19T^{2} \) |
| 23 | \( 1 - 2.14iT - 23T^{2} \) |
| 29 | \( 1 + 1.98iT - 29T^{2} \) |
| 31 | \( 1 - 2.86iT - 31T^{2} \) |
| 37 | \( 1 + 0.233T + 37T^{2} \) |
| 41 | \( 1 - 1.78T + 41T^{2} \) |
| 43 | \( 1 + 9.77iT - 43T^{2} \) |
| 47 | \( 1 + 5.43iT - 47T^{2} \) |
| 53 | \( 1 - 5.11T + 53T^{2} \) |
| 59 | \( 1 - 14.9iT - 59T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 71 | \( 1 - 9.67iT - 71T^{2} \) |
| 73 | \( 1 + 9.52T + 73T^{2} \) |
| 79 | \( 1 + 5.92iT - 79T^{2} \) |
| 83 | \( 1 - 5.42iT - 83T^{2} \) |
| 89 | \( 1 + 3.00iT - 89T^{2} \) |
| 97 | \( 1 + 11.2iT - 97T^{2} \) |
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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
−8.360504164907825069648266902334, −7.25054626416310190910397113350, −7.15916479101665100716201633817, −5.87269198310733716676027793203, −5.64653193105323905135367344965, −4.31895692936239263074547722650, −3.65990347056175488128159336154, −2.85030129828917336026422657761, −1.68897584199613603488442659869, −0.831925452999845742382132231679,
0.61330382362001489664179291792, 1.53307523962305345393139594519, 2.65299166696568579144678559320, 3.48246038095189275638957275671, 4.35480402758117036191314778204, 5.07260581133815785569049415423, 6.15495999901556644904844033889, 6.76128515485560416392789580856, 7.37534837231078324880568485050, 7.941320876923277705154314746914