| L(s) = 1 | + 3.07·3-s + 1.60·5-s − 4.23·7-s + 6.43·9-s − 0.394·11-s + 0.612·13-s + 4.93·15-s − 7.39·17-s − 6.08·19-s − 13.0·21-s − 4.34·23-s − 2.42·25-s + 10.5·27-s − 7.97·29-s + 4.45·31-s − 1.21·33-s − 6.80·35-s + 3.92·37-s + 1.88·39-s + 5.78·43-s + 10.3·45-s − 1.45·47-s + 10.9·49-s − 22.7·51-s − 8.34·53-s − 0.633·55-s − 18.6·57-s + ⋯ |
| L(s) = 1 | + 1.77·3-s + 0.718·5-s − 1.60·7-s + 2.14·9-s − 0.118·11-s + 0.169·13-s + 1.27·15-s − 1.79·17-s − 1.39·19-s − 2.84·21-s − 0.905·23-s − 0.484·25-s + 2.03·27-s − 1.48·29-s + 0.800·31-s − 0.211·33-s − 1.15·35-s + 0.645·37-s + 0.301·39-s + 0.882·43-s + 1.54·45-s − 0.212·47-s + 1.56·49-s − 3.17·51-s − 1.14·53-s − 0.0854·55-s − 2.47·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 - 3.07T + 3T^{2} \) |
| 5 | \( 1 - 1.60T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 0.394T + 11T^{2} \) |
| 13 | \( 1 - 0.612T + 13T^{2} \) |
| 17 | \( 1 + 7.39T + 17T^{2} \) |
| 19 | \( 1 + 6.08T + 19T^{2} \) |
| 23 | \( 1 + 4.34T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 - 4.45T + 31T^{2} \) |
| 37 | \( 1 - 3.92T + 37T^{2} \) |
| 43 | \( 1 - 5.78T + 43T^{2} \) |
| 47 | \( 1 + 1.45T + 47T^{2} \) |
| 53 | \( 1 + 8.34T + 53T^{2} \) |
| 59 | \( 1 - 0.682T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 1.18T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 9.49T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
| 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
−7.76388092296597706634496847763, −6.87014198269177046708985955263, −6.43555962478589543396252036660, −5.72949463494353113206670042144, −4.20687813362422462965962325299, −4.03653656965098129035627584811, −2.97317135240790093605317021984, −2.38747160159218763530384397225, −1.82293423022666886405785987764, 0,
1.82293423022666886405785987764, 2.38747160159218763530384397225, 2.97317135240790093605317021984, 4.03653656965098129035627584811, 4.20687813362422462965962325299, 5.72949463494353113206670042144, 6.43555962478589543396252036660, 6.87014198269177046708985955263, 7.76388092296597706634496847763
