| L(s) = 1 | − 11-s − 2·13-s − 2·17-s + 6·19-s − 4·23-s − 5·25-s + 10·29-s − 8·31-s + 6·37-s − 6·41-s + 6·43-s − 12·53-s + 12·59-s − 6·61-s + 8·67-s − 8·71-s + 10·73-s − 10·79-s + 4·83-s + 16·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s − 0.554·13-s − 0.485·17-s + 1.37·19-s − 0.834·23-s − 25-s + 1.85·29-s − 1.43·31-s + 0.986·37-s − 0.937·41-s + 0.914·43-s − 1.64·53-s + 1.56·59-s − 0.768·61-s + 0.977·67-s − 0.949·71-s + 1.17·73-s − 1.12·79-s + 0.439·83-s + 1.69·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.970795704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.970795704\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.67200268015365, −12.12918565422608, −11.72576091016533, −11.43592665710914, −10.79604931031291, −10.33385231838943, −9.897840552005124, −9.467874633381657, −9.125923523242978, −8.375518523148031, −8.055749922440113, −7.509306431975363, −7.210137088259726, −6.526879210598596, −6.086468690092777, −5.575952815109843, −5.049815715676627, −4.600835031873075, −4.067123401176268, −3.387619300934343, −3.016841879230715, −2.229677485454282, −1.921803513533608, −1.029400654835726, −0.4075353571859824,
0.4075353571859824, 1.029400654835726, 1.921803513533608, 2.229677485454282, 3.016841879230715, 3.387619300934343, 4.067123401176268, 4.600835031873075, 5.049815715676627, 5.575952815109843, 6.086468690092777, 6.526879210598596, 7.210137088259726, 7.509306431975363, 8.055749922440113, 8.375518523148031, 9.125923523242978, 9.467874633381657, 9.897840552005124, 10.33385231838943, 10.79604931031291, 11.43592665710914, 11.72576091016533, 12.12918565422608, 12.67200268015365
