| L(s) = 1 | + 2·3-s − 14·5-s − 23·9-s + 20·11-s + 6·13-s − 28·15-s − 20·17-s + 102·19-s + 124·23-s + 71·25-s − 100·27-s − 78·29-s + 236·31-s + 40·33-s + 66·37-s + 12·39-s − 268·41-s − 132·43-s + 322·45-s + 516·47-s − 40·51-s − 354·53-s − 280·55-s + 204·57-s + 438·59-s − 486·61-s − 84·65-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 1.25·5-s − 0.851·9-s + 0.548·11-s + 0.128·13-s − 0.481·15-s − 0.285·17-s + 1.23·19-s + 1.12·23-s + 0.567·25-s − 0.712·27-s − 0.499·29-s + 1.36·31-s + 0.211·33-s + 0.293·37-s + 0.0492·39-s − 1.02·41-s − 0.468·43-s + 1.06·45-s + 1.60·47-s − 0.109·51-s − 0.917·53-s − 0.686·55-s + 0.474·57-s + 0.966·59-s − 1.02·61-s − 0.160·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 20 T + p^{3} T^{2} \) |
| 19 | \( 1 - 102 T + p^{3} T^{2} \) |
| 23 | \( 1 - 124 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 236 T + p^{3} T^{2} \) |
| 37 | \( 1 - 66 T + p^{3} T^{2} \) |
| 41 | \( 1 + 268 T + p^{3} T^{2} \) |
| 43 | \( 1 + 132 T + p^{3} T^{2} \) |
| 47 | \( 1 - 516 T + p^{3} T^{2} \) |
| 53 | \( 1 + 354 T + p^{3} T^{2} \) |
| 59 | \( 1 - 438 T + p^{3} T^{2} \) |
| 61 | \( 1 + 486 T + p^{3} T^{2} \) |
| 67 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 71 | \( 1 + 248 T + p^{3} T^{2} \) |
| 73 | \( 1 + 768 T + p^{3} T^{2} \) |
| 79 | \( 1 + 192 T + p^{3} T^{2} \) |
| 83 | \( 1 - 294 T + p^{3} T^{2} \) |
| 89 | \( 1 + 80 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1404 T + p^{3} T^{2} \) |
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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
−8.702630558602801179991529491676, −7.85480618472471378533358161768, −7.28388053231733005671836599343, −6.31299710447616097188335390881, −5.27436368011111061424909288785, −4.32335288206824020520575311687, −3.43513761531655175734344357525, −2.77114143500530231589517177753, −1.19365305107458053492591488706, 0,
1.19365305107458053492591488706, 2.77114143500530231589517177753, 3.43513761531655175734344357525, 4.32335288206824020520575311687, 5.27436368011111061424909288785, 6.31299710447616097188335390881, 7.28388053231733005671836599343, 7.85480618472471378533358161768, 8.702630558602801179991529491676
