| L(s) = 1 | + 2-s + 6·3-s − 7·4-s − 5·5-s + 6·6-s − 15·8-s + 9·9-s − 5·10-s − 44·11-s − 42·12-s + 6·13-s − 30·15-s + 41·16-s − 24·17-s + 9·18-s − 114·19-s + 35·20-s − 44·22-s − 52·23-s − 90·24-s + 25·25-s + 6·26-s − 108·27-s + 146·29-s − 30·30-s − 276·31-s + 161·32-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 1.15·3-s − 7/8·4-s − 0.447·5-s + 0.408·6-s − 0.662·8-s + 1/3·9-s − 0.158·10-s − 1.20·11-s − 1.01·12-s + 0.128·13-s − 0.516·15-s + 0.640·16-s − 0.342·17-s + 0.117·18-s − 1.37·19-s + 0.391·20-s − 0.426·22-s − 0.471·23-s − 0.765·24-s + 1/5·25-s + 0.0452·26-s − 0.769·27-s + 0.934·29-s − 0.182·30-s − 1.59·31-s + 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 24 T + p^{3} T^{2} \) |
| 19 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 - 146 T + p^{3} T^{2} \) |
| 31 | \( 1 + 276 T + p^{3} T^{2} \) |
| 37 | \( 1 + 210 T + p^{3} T^{2} \) |
| 41 | \( 1 - 444 T + p^{3} T^{2} \) |
| 43 | \( 1 - 492 T + p^{3} T^{2} \) |
| 47 | \( 1 + 612 T + p^{3} T^{2} \) |
| 53 | \( 1 - 50 T + p^{3} T^{2} \) |
| 59 | \( 1 - 294 T + p^{3} T^{2} \) |
| 61 | \( 1 - 450 T + p^{3} T^{2} \) |
| 67 | \( 1 + 668 T + p^{3} T^{2} \) |
| 71 | \( 1 + 308 T + p^{3} T^{2} \) |
| 73 | \( 1 - 12 T + p^{3} T^{2} \) |
| 79 | \( 1 - 596 T + p^{3} T^{2} \) |
| 83 | \( 1 + 966 T + p^{3} T^{2} \) |
| 89 | \( 1 + 408 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1200 T + p^{3} T^{2} \) |
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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
−11.11943207937774431517957557187, −10.07079604666857719509341197608, −8.930214853448494380417179699339, −8.394596957009384754322153936415, −7.50764558697864415862005856022, −5.85972375663024862651738899135, −4.58125118838758732899166915410, −3.59686616752362928862528365883, −2.42548739097642481039152469598, 0,
2.42548739097642481039152469598, 3.59686616752362928862528365883, 4.58125118838758732899166915410, 5.85972375663024862651738899135, 7.50764558697864415862005856022, 8.394596957009384754322153936415, 8.930214853448494380417179699339, 10.07079604666857719509341197608, 11.11943207937774431517957557187
