| L(s) = 1 | − 1.47·2-s + 0.170·4-s + 3.86·7-s + 2.69·8-s − 0.260·11-s − 4.07·13-s − 5.69·14-s − 4.31·16-s − 3.26·17-s + 4.24·19-s + 0.383·22-s − 8.69·23-s + 6.00·26-s + 0.659·28-s − 4.22·29-s + 2.65·31-s + 0.961·32-s + 4.80·34-s − 2.27·37-s − 6.26·38-s − 5.64·41-s − 9.07·43-s − 0.0443·44-s + 12.8·46-s + 1.42·47-s + 7.94·49-s − 0.695·52-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.0852·4-s + 1.46·7-s + 0.952·8-s − 0.0784·11-s − 1.13·13-s − 1.52·14-s − 1.07·16-s − 0.790·17-s + 0.974·19-s + 0.0817·22-s − 1.81·23-s + 1.17·26-s + 0.124·28-s − 0.784·29-s + 0.476·31-s + 0.170·32-s + 0.823·34-s − 0.374·37-s − 1.01·38-s − 0.881·41-s − 1.38·43-s − 0.00668·44-s + 1.88·46-s + 0.208·47-s + 1.13·49-s − 0.0964·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.47T + 2T^{2} \) |
| 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 + 0.260T + 11T^{2} \) |
| 13 | \( 1 + 4.07T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 8.69T + 23T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 - 2.65T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 + 5.64T + 41T^{2} \) |
| 43 | \( 1 + 9.07T + 43T^{2} \) |
| 47 | \( 1 - 1.42T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 - 2.52T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 8.38T + 71T^{2} \) |
| 73 | \( 1 - 0.403T + 73T^{2} \) |
| 79 | \( 1 + 3.04T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 + 7.17T + 89T^{2} \) |
| 97 | \( 1 - 3.11T + 97T^{2} \) |
| 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
−8.596462851604463335111333849164, −8.091230196508658538285130038172, −7.55003984790632294327289400159, −6.71489873929404732546534868617, −5.32082535114291562647040008944, −4.82228336415293173970280906721, −3.90198070562548266410989640944, −2.27166334910496935078208113591, −1.51986526300393663371883431814, 0,
1.51986526300393663371883431814, 2.27166334910496935078208113591, 3.90198070562548266410989640944, 4.82228336415293173970280906721, 5.32082535114291562647040008944, 6.71489873929404732546534868617, 7.55003984790632294327289400159, 8.091230196508658538285130038172, 8.596462851604463335111333849164
