| L(s) = 1 | − 0.362·2-s − 3.41·3-s − 1.86·4-s + 1.23·6-s − 3.34·7-s + 1.40·8-s + 8.66·9-s − 2.67·11-s + 6.38·12-s − 0.591·13-s + 1.21·14-s + 3.22·16-s + 1.07·17-s − 3.14·18-s − 5.88·19-s + 11.4·21-s + 0.971·22-s − 2.98·23-s − 4.79·24-s + 0.214·26-s − 19.3·27-s + 6.24·28-s + 3.96·29-s − 4.71·31-s − 3.97·32-s + 9.14·33-s − 0.390·34-s + ⋯ |
| L(s) = 1 | − 0.256·2-s − 1.97·3-s − 0.934·4-s + 0.505·6-s − 1.26·7-s + 0.496·8-s + 2.88·9-s − 0.807·11-s + 1.84·12-s − 0.163·13-s + 0.323·14-s + 0.806·16-s + 0.261·17-s − 0.741·18-s − 1.35·19-s + 2.48·21-s + 0.207·22-s − 0.621·23-s − 0.978·24-s + 0.0420·26-s − 3.72·27-s + 1.17·28-s + 0.737·29-s − 0.846·31-s − 0.703·32-s + 1.59·33-s − 0.0670·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 0.362T + 2T^{2} \) |
| 3 | \( 1 + 3.41T + 3T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 13 | \( 1 + 0.591T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 + 2.98T + 23T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 + 4.71T + 31T^{2} \) |
| 37 | \( 1 + 8.86T + 37T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 - 2.42T + 43T^{2} \) |
| 47 | \( 1 - 9.02T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 - 9.68T + 59T^{2} \) |
| 61 | \( 1 + 0.830T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 0.250T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 7.04T + 83T^{2} \) |
| 89 | \( 1 - 6.66T + 89T^{2} \) |
| 97 | \( 1 + 0.0770T + 97T^{2} \) |
| 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
−7.51277352774704783134786612741, −6.85283433038703534211874714108, −6.24181873386689973831523375539, −5.50378389585711596901651385255, −5.07969970768654819723829419256, −4.19125973540422332138788656026, −3.59086862130365589723009739816, −2.03409685923488021915930703541, −0.67811407186717908299955099431, 0,
0.67811407186717908299955099431, 2.03409685923488021915930703541, 3.59086862130365589723009739816, 4.19125973540422332138788656026, 5.07969970768654819723829419256, 5.50378389585711596901651385255, 6.24181873386689973831523375539, 6.85283433038703534211874714108, 7.51277352774704783134786612741
