| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s − 24·7-s + 8·8-s + 10·10-s + 32·11-s + 13·13-s − 48·14-s + 16·16-s − 78·17-s − 32·19-s + 20·20-s + 64·22-s − 158·23-s + 25·25-s + 26·26-s − 96·28-s − 126·29-s + 250·31-s + 32·32-s − 156·34-s − 120·35-s + 38·37-s − 64·38-s + 40·40-s − 90·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.29·7-s + 0.353·8-s + 0.316·10-s + 0.877·11-s + 0.277·13-s − 0.916·14-s + 1/4·16-s − 1.11·17-s − 0.386·19-s + 0.223·20-s + 0.620·22-s − 1.43·23-s + 1/5·25-s + 0.196·26-s − 0.647·28-s − 0.806·29-s + 1.44·31-s + 0.176·32-s − 0.786·34-s − 0.579·35-s + 0.168·37-s − 0.273·38-s + 0.158·40-s − 0.342·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 32 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 32 T + p^{3} T^{2} \) |
| 23 | \( 1 + 158 T + p^{3} T^{2} \) |
| 29 | \( 1 + 126 T + p^{3} T^{2} \) |
| 31 | \( 1 - 250 T + p^{3} T^{2} \) |
| 37 | \( 1 - 38 T + p^{3} T^{2} \) |
| 41 | \( 1 + 90 T + p^{3} T^{2} \) |
| 43 | \( 1 + 428 T + p^{3} T^{2} \) |
| 47 | \( 1 - 84 T + p^{3} T^{2} \) |
| 53 | \( 1 + 236 T + p^{3} T^{2} \) |
| 59 | \( 1 - 188 T + p^{3} T^{2} \) |
| 61 | \( 1 + 170 T + p^{3} T^{2} \) |
| 67 | \( 1 + 78 T + p^{3} T^{2} \) |
| 71 | \( 1 + 16 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1012 T + p^{3} T^{2} \) |
| 79 | \( 1 + 16 p T + p^{3} T^{2} \) |
| 83 | \( 1 - 60 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1010 T + p^{3} T^{2} \) |
| 97 | \( 1 - 372 T + p^{3} T^{2} \) |
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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
−9.121279223649185586386626270498, −8.201913402516203246774975732630, −6.88845762720820349347712946630, −6.41271810586749764311972996003, −5.80530936665485637456309941288, −4.50596107460498576066103106824, −3.74949914487496881264070984712, −2.75631234700538252266863372668, −1.65662243114137216619605093476, 0,
1.65662243114137216619605093476, 2.75631234700538252266863372668, 3.74949914487496881264070984712, 4.50596107460498576066103106824, 5.80530936665485637456309941288, 6.41271810586749764311972996003, 6.88845762720820349347712946630, 8.201913402516203246774975732630, 9.121279223649185586386626270498
