| L(s) = 1 | − 2-s + 4-s − 1.77·5-s + 0.833·7-s − 8-s + 1.77·10-s − 3.92·11-s + 4.58·13-s − 0.833·14-s + 16-s − 1.16·17-s − 3.74·19-s − 1.77·20-s + 3.92·22-s − 1.83·25-s − 4.58·26-s + 0.833·28-s − 1.08·29-s + 7.05·31-s − 32-s + 1.16·34-s − 1.48·35-s + 5.44·37-s + 3.74·38-s + 1.77·40-s + 1.45·41-s + 3.91·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.795·5-s + 0.315·7-s − 0.353·8-s + 0.562·10-s − 1.18·11-s + 1.27·13-s − 0.222·14-s + 0.250·16-s − 0.282·17-s − 0.859·19-s − 0.397·20-s + 0.836·22-s − 0.366·25-s − 0.899·26-s + 0.157·28-s − 0.202·29-s + 1.26·31-s − 0.176·32-s + 0.200·34-s − 0.250·35-s + 0.895·37-s + 0.607·38-s + 0.281·40-s + 0.226·41-s + 0.597·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 1.77T + 5T^{2} \) |
| 7 | \( 1 - 0.833T + 7T^{2} \) |
| 11 | \( 1 + 3.92T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 + 1.16T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 29 | \( 1 + 1.08T + 29T^{2} \) |
| 31 | \( 1 - 7.05T + 31T^{2} \) |
| 37 | \( 1 - 5.44T + 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 - 3.91T + 43T^{2} \) |
| 47 | \( 1 - 5.02T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 3.91T + 59T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 - 2.05T + 67T^{2} \) |
| 71 | \( 1 - 7.83T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 9.56T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 2.07T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.79395361922571139216281073924, −6.67886040716189126232655933476, −6.19699314999521741695159086263, −5.35197866541089343842473937893, −4.47034498481190090501929294885, −3.82649181621948128350791690649, −2.90481312244896931641191073993, −2.12915690657576982274498336654, −1.03676310587846072532187939824, 0,
1.03676310587846072532187939824, 2.12915690657576982274498336654, 2.90481312244896931641191073993, 3.82649181621948128350791690649, 4.47034498481190090501929294885, 5.35197866541089343842473937893, 6.19699314999521741695159086263, 6.67886040716189126232655933476, 7.79395361922571139216281073924
