| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 15-s + 16-s + 18-s − 20-s + 23-s − 24-s + 25-s − 27-s + 30-s + 32-s + 36-s − 40-s − 45-s + 46-s − 48-s + 49-s + 50-s + 2·53-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 15-s + 16-s + 18-s − 20-s + 23-s − 24-s + 25-s − 27-s + 30-s + 32-s + 36-s − 40-s − 45-s + 46-s − 48-s + 49-s + 50-s + 2·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.529219989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.529219989\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 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
−8.914984996567196365886147990882, −7.950017631138215629064614923180, −7.10671666257944183970594909715, −6.79719965123492992921503307345, −5.71356855095690665633157219651, −5.13023633444683593155241847107, −4.28895602170118283425453180554, −3.70397519229352340275897967610, −2.56383274867056733123772112020, −1.08874900475449945545404627057,
1.08874900475449945545404627057, 2.56383274867056733123772112020, 3.70397519229352340275897967610, 4.28895602170118283425453180554, 5.13023633444683593155241847107, 5.71356855095690665633157219651, 6.79719965123492992921503307345, 7.10671666257944183970594909715, 7.950017631138215629064614923180, 8.914984996567196365886147990882
