| L(s) = 1 | + 2-s + 4-s + 1.41·7-s + 8-s − 1.41·11-s + 1.41·14-s + 16-s − 1.41·19-s − 1.41·22-s − 25-s + 1.41·28-s + 32-s − 1.41·38-s − 41-s − 1.41·44-s + 1.41·47-s + 1.00·49-s − 50-s + 1.41·56-s − 2·61-s + 64-s + 1.41·67-s + 1.41·71-s − 1.41·76-s − 2.00·77-s − 1.41·79-s − 82-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 1.41·7-s + 8-s − 1.41·11-s + 1.41·14-s + 16-s − 1.41·19-s − 1.41·22-s − 25-s + 1.41·28-s + 32-s − 1.41·38-s − 41-s − 1.41·44-s + 1.41·47-s + 1.00·49-s − 50-s + 1.41·56-s − 2·61-s + 64-s + 1.41·67-s + 1.41·71-s − 1.41·76-s − 2.00·77-s − 1.41·79-s − 82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.166819926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.166819926\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−10.01421226632330594889713470662, −8.576248010243203570525885907267, −7.956604582039022768788725691215, −7.32973813055898425318770593469, −6.21390237571841117597290264831, −5.36805907639702349566274186492, −4.74543900634871059464864485533, −3.91819557144235609080736023061, −2.58434719491712651022413391077, −1.80455648630118609834177336532,
1.80455648630118609834177336532, 2.58434719491712651022413391077, 3.91819557144235609080736023061, 4.74543900634871059464864485533, 5.36805907639702349566274186492, 6.21390237571841117597290264831, 7.32973813055898425318770593469, 7.956604582039022768788725691215, 8.576248010243203570525885907267, 10.01421226632330594889713470662
