| L(s) = 1 | − 3·2-s + 4·3-s + 4-s + 12·5-s − 12·6-s − 7·7-s + 21·8-s − 11·9-s − 36·10-s + 4·12-s − 38·13-s + 21·14-s + 48·15-s − 71·16-s + 48·17-s + 33·18-s + 70·19-s + 12·20-s − 28·21-s + 12·23-s + 84·24-s + 19·25-s + 114·26-s − 152·27-s − 7·28-s − 126·29-s − 144·30-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.769·3-s + 1/8·4-s + 1.07·5-s − 0.816·6-s − 0.377·7-s + 0.928·8-s − 0.407·9-s − 1.13·10-s + 0.0962·12-s − 0.810·13-s + 0.400·14-s + 0.826·15-s − 1.10·16-s + 0.684·17-s + 0.432·18-s + 0.845·19-s + 0.134·20-s − 0.290·21-s + 0.108·23-s + 0.714·24-s + 0.151·25-s + 0.859·26-s − 1.08·27-s − 0.0472·28-s − 0.806·29-s − 0.876·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + p T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 48 T + p^{3} T^{2} \) |
| 19 | \( 1 - 70 T + p^{3} T^{2} \) |
| 23 | \( 1 - 12 T + p^{3} T^{2} \) |
| 29 | \( 1 + 126 T + p^{3} T^{2} \) |
| 31 | \( 1 + 70 T + p^{3} T^{2} \) |
| 37 | \( 1 + 358 T + p^{3} T^{2} \) |
| 41 | \( 1 - 216 T + p^{3} T^{2} \) |
| 43 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 390 T + p^{3} T^{2} \) |
| 53 | \( 1 - 438 T + p^{3} T^{2} \) |
| 59 | \( 1 + 552 T + p^{3} T^{2} \) |
| 61 | \( 1 + 830 T + p^{3} T^{2} \) |
| 67 | \( 1 + 196 T + p^{3} T^{2} \) |
| 71 | \( 1 - 648 T + p^{3} T^{2} \) |
| 73 | \( 1 - 16 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1352 T + p^{3} T^{2} \) |
| 83 | \( 1 + 90 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 + 70 T + p^{3} T^{2} \) |
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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
−9.263264549601838363045027285268, −8.883774208926519972699574652036, −7.78152934434122044752447215896, −7.21933793918777285262745124545, −5.88694145043051672375347517637, −5.07709988116926930587471171569, −3.58141223781575115856465740341, −2.47206495890642451598397643915, −1.49604714505811198885013045503, 0,
1.49604714505811198885013045503, 2.47206495890642451598397643915, 3.58141223781575115856465740341, 5.07709988116926930587471171569, 5.88694145043051672375347517637, 7.21933793918777285262745124545, 7.78152934434122044752447215896, 8.883774208926519972699574652036, 9.263264549601838363045027285268
