| L(s) = 1 | − 1.55·2-s + 3-s + 0.409·4-s − 1.55·6-s + 2.46·8-s + 9-s + 4.43·11-s + 0.409·12-s + 1.73·13-s − 4.65·16-s − 2.73·17-s − 1.55·18-s − 0.305·19-s − 6.87·22-s − 7.02·23-s + 2.46·24-s − 2.68·26-s + 27-s − 7.79·29-s − 5.28·31-s + 2.28·32-s + 4.43·33-s + 4.24·34-s + 0.409·36-s − 3.67·37-s + 0.473·38-s + 1.73·39-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.577·3-s + 0.204·4-s − 0.633·6-s + 0.872·8-s + 0.333·9-s + 1.33·11-s + 0.118·12-s + 0.480·13-s − 1.16·16-s − 0.662·17-s − 0.365·18-s − 0.0700·19-s − 1.46·22-s − 1.46·23-s + 0.503·24-s − 0.527·26-s + 0.192·27-s − 1.44·29-s − 0.949·31-s + 0.403·32-s + 0.771·33-s + 0.727·34-s + 0.0683·36-s − 0.604·37-s + 0.0768·38-s + 0.277·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 + 0.305T + 19T^{2} \) |
| 23 | \( 1 + 7.02T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 - 1.81T + 47T^{2} \) |
| 53 | \( 1 + 1.71T + 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 - 9.55T + 61T^{2} \) |
| 67 | \( 1 + 8.38T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 - 5.09T + 83T^{2} \) |
| 89 | \( 1 - 4.07T + 89T^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−8.433498245303075707547341918450, −7.61545483644683386698549111932, −6.92611159594906509340753462844, −6.21241636935230905127244747441, −5.08448624212120699747763168331, −4.02105416149781718471520502500, −3.61823541569508247029507250307, −2.03819045241024911082761473782, −1.48873476158269063007092415890, 0,
1.48873476158269063007092415890, 2.03819045241024911082761473782, 3.61823541569508247029507250307, 4.02105416149781718471520502500, 5.08448624212120699747763168331, 6.21241636935230905127244747441, 6.92611159594906509340753462844, 7.61545483644683386698549111932, 8.433498245303075707547341918450
