| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.557404813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557404813\) |
| \(L(1)\) |
|
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
−11.14478443925547576013609649673, −9.878191527030714263891319301318, −9.203490893724311193910410343899, −8.030617649835787760899180467467, −6.82428609521709537247907988016, −6.24032611945626527976617693619, −5.45733683174825137938095995949, −3.93468807116862272213773832284, −3.47086049052007197481262615160, −1.98310995698250958628581129572,
1.98310995698250958628581129572, 3.47086049052007197481262615160, 3.93468807116862272213773832284, 5.45733683174825137938095995949, 6.24032611945626527976617693619, 6.82428609521709537247907988016, 8.030617649835787760899180467467, 9.203490893724311193910410343899, 9.878191527030714263891319301318, 11.14478443925547576013609649673
