L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 9-s − 2·11-s + 14-s + 16-s − 18-s − 2·22-s + 23-s − 25-s + 28-s − 2·29-s + 32-s − 36-s + 2·43-s − 2·44-s + 46-s + 49-s − 50-s + 56-s − 2·58-s − 63-s + 64-s − 2·67-s − 72-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 9-s − 2·11-s + 14-s + 16-s − 18-s − 2·22-s + 23-s − 25-s + 28-s − 2·29-s + 32-s − 36-s + 2·43-s − 2·44-s + 46-s + 49-s − 50-s + 56-s − 2·58-s − 63-s + 64-s − 2·67-s − 72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.612983706\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.612983706\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( ( 1 + T )^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.06669482710288292267270479227, −10.30549427916904613439695298262, −8.898546386816330238296533487589, −7.79127399376291677453609331731, −7.44736227839790996096202827762, −5.75746833839907500924330598764, −5.44110125691855176546792568015, −4.41286652103000778035860061174, −3.04923109203944963433407500630, −2.12095819084743653981309789903,
2.12095819084743653981309789903, 3.04923109203944963433407500630, 4.41286652103000778035860061174, 5.44110125691855176546792568015, 5.75746833839907500924330598764, 7.44736227839790996096202827762, 7.79127399376291677453609331731, 8.898546386816330238296533487589, 10.30549427916904613439695298262, 11.06669482710288292267270479227