| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s + 9-s − 10-s − 11-s − 13-s − 14-s + 16-s + 18-s − 20-s − 22-s − 23-s − 26-s − 28-s + 2·31-s + 32-s + 35-s + 36-s − 40-s − 41-s + 2·43-s − 44-s − 45-s − 46-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s + 9-s − 10-s − 11-s − 13-s − 14-s + 16-s + 18-s − 20-s − 22-s − 23-s − 26-s − 28-s + 2·31-s + 32-s + 35-s + 36-s − 40-s − 41-s + 2·43-s − 44-s − 45-s − 46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.055350246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.055350246\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−12.40859736833047858016597362994, −11.77010407140383767620778671010, −10.45481377185261755189014576075, −9.831509504140551234548252796189, −7.969950791834566630990876706587, −7.29655597379814193998474844706, −6.24928589640252769920597090656, −4.83876716620938424638612762836, −3.90413047644985056499535338872, −2.64366530371542938651687617004,
2.64366530371542938651687617004, 3.90413047644985056499535338872, 4.83876716620938424638612762836, 6.24928589640252769920597090656, 7.29655597379814193998474844706, 7.969950791834566630990876706587, 9.831509504140551234548252796189, 10.45481377185261755189014576075, 11.77010407140383767620778671010, 12.40859736833047858016597362994
