| L(s) = 1 | + 2.82·3-s + 2.82·5-s + 5.00·9-s − 4·11-s − 2.82·13-s + 8.00·15-s − 5.65·17-s − 2.82·19-s + 3.00·25-s + 5.65·27-s + 2·29-s − 5.65·31-s − 11.3·33-s + 10·37-s − 8.00·39-s + 5.65·41-s − 4·43-s + 14.1·45-s + 5.65·47-s − 16.0·51-s + 6·53-s − 11.3·55-s − 8.00·57-s − 2.82·59-s + 14.1·61-s − 8.00·65-s + 12·67-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 1.26·5-s + 1.66·9-s − 1.20·11-s − 0.784·13-s + 2.06·15-s − 1.37·17-s − 0.648·19-s + 0.600·25-s + 1.08·27-s + 0.371·29-s − 1.01·31-s − 1.96·33-s + 1.64·37-s − 1.28·39-s + 0.883·41-s − 0.609·43-s + 2.10·45-s + 0.825·47-s − 2.24·51-s + 0.824·53-s − 1.52·55-s − 1.05·57-s − 0.368·59-s + 1.81·61-s − 0.992·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.455602122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.455602122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 5.65T + 97T^{2} \) |
| 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
−11.02003112888352876795354260785, −10.06505285963591712916076259719, −9.445249646624673432647354616810, −8.648320943661838632975628604705, −7.77258515529897134687955321671, −6.75192047741729539144599946478, −5.44719222554173891922828470993, −4.21172775218603880989909672935, −2.58900169588139173158210362787, −2.19527274733034608976408563982,
2.19527274733034608976408563982, 2.58900169588139173158210362787, 4.21172775218603880989909672935, 5.44719222554173891922828470993, 6.75192047741729539144599946478, 7.77258515529897134687955321671, 8.648320943661838632975628604705, 9.445249646624673432647354616810, 10.06505285963591712916076259719, 11.02003112888352876795354260785
