| L(s) = 1 | − 3-s + 9-s + 2·11-s − 3·17-s + 8·19-s − 7·23-s − 27-s − 2·29-s − 31-s − 2·33-s − 3·41-s + 8·43-s + 3·47-s + 3·51-s + 12·53-s − 8·57-s − 10·59-s − 2·61-s − 4·67-s + 7·69-s − 3·71-s − 2·73-s − 13·79-s + 81-s + 6·83-s + 2·87-s − 13·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.727·17-s + 1.83·19-s − 1.45·23-s − 0.192·27-s − 0.371·29-s − 0.179·31-s − 0.348·33-s − 0.468·41-s + 1.21·43-s + 0.437·47-s + 0.420·51-s + 1.64·53-s − 1.05·57-s − 1.30·59-s − 0.256·61-s − 0.488·67-s + 0.842·69-s − 0.356·71-s − 0.234·73-s − 1.46·79-s + 1/9·81-s + 0.658·83-s + 0.214·87-s − 1.37·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.87568580669258, −13.35466378011667, −12.86275544380414, −12.17852242981210, −11.82939515939878, −11.61920133896993, −10.90925874655133, −10.49450520868943, −9.915815043259889, −9.476602260566418, −9.010490968684737, −8.476825339853128, −7.715595677851434, −7.371977625736020, −6.898902141734362, −6.198363574016130, −5.773455342104529, −5.390394652237799, −4.587916174080013, −4.186362270638373, −3.594300447857252, −2.939660494306464, −2.167224567001075, −1.517835619568320, −0.8456759066572029, 0,
0.8456759066572029, 1.517835619568320, 2.167224567001075, 2.939660494306464, 3.594300447857252, 4.186362270638373, 4.587916174080013, 5.390394652237799, 5.773455342104529, 6.198363574016130, 6.898902141734362, 7.371977625736020, 7.715595677851434, 8.476825339853128, 9.010490968684737, 9.476602260566418, 9.915815043259889, 10.49450520868943, 10.90925874655133, 11.61920133896993, 11.82939515939878, 12.17852242981210, 12.86275544380414, 13.35466378011667, 13.87568580669258
