| L(s) = 1 | − 1.41·2-s − 2.44·7-s + 2.82·8-s − 1.73·11-s + 4.89·13-s + 3.46·14-s − 4.00·16-s − 1.41·17-s − 5·19-s + 2.44·22-s + 5.65·23-s − 6.92·26-s + 1.73·29-s − 7·31-s + 2.00·34-s + 7.34·37-s + 7.07·38-s + 12.1·41-s − 12.2·43-s − 8.00·46-s + 7.07·47-s − 1.00·49-s + 1.41·53-s − 6.92·56-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.925·7-s + 0.999·8-s − 0.522·11-s + 1.35·13-s + 0.925·14-s − 1.00·16-s − 0.342·17-s − 1.14·19-s + 0.522·22-s + 1.17·23-s − 1.35·26-s + 0.321·29-s − 1.25·31-s + 0.342·34-s + 1.20·37-s + 1.14·38-s + 1.89·41-s − 1.86·43-s − 1.17·46-s + 1.03·47-s − 0.142·49-s + 0.194·53-s − 0.925·56-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 | \( 1 + 1.41T + 2T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 7.34T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 4.89T + 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 - 7.34T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 2.44T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.866925373486016565968953608182, −8.203708392622980327778776907399, −7.33130735054095861571617972663, −6.53870675302243277160333666939, −5.75310444993113372663820050663, −4.58944468630423561909515536022, −3.74856321248017809122355846391, −2.61694453210578327853079755066, −1.28022409683729145751209056930, 0,
1.28022409683729145751209056930, 2.61694453210578327853079755066, 3.74856321248017809122355846391, 4.58944468630423561909515536022, 5.75310444993113372663820050663, 6.53870675302243277160333666939, 7.33130735054095861571617972663, 8.203708392622980327778776907399, 8.866925373486016565968953608182
