| L(s) = 1 | + 2-s + 3-s + 4-s − 0.684·5-s + 6-s − 1.46·7-s + 8-s + 9-s − 0.684·10-s − 0.852·11-s + 12-s − 13-s − 1.46·14-s − 0.684·15-s + 16-s + 4.18·17-s + 18-s − 7.25·19-s − 0.684·20-s − 1.46·21-s − 0.852·22-s + 8.93·23-s + 24-s − 4.53·25-s − 26-s + 27-s − 1.46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.306·5-s + 0.408·6-s − 0.553·7-s + 0.353·8-s + 0.333·9-s − 0.216·10-s − 0.256·11-s + 0.288·12-s − 0.277·13-s − 0.391·14-s − 0.176·15-s + 0.250·16-s + 1.01·17-s + 0.235·18-s − 1.66·19-s − 0.153·20-s − 0.319·21-s − 0.181·22-s + 1.86·23-s + 0.204·24-s − 0.906·25-s − 0.196·26-s + 0.192·27-s − 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 0.684T + 5T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 + 0.852T + 11T^{2} \) |
| 17 | \( 1 - 4.18T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 - 8.93T + 23T^{2} \) |
| 29 | \( 1 + 9.35T + 29T^{2} \) |
| 31 | \( 1 + 7.80T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 - 7.52T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 6.03T + 47T^{2} \) |
| 53 | \( 1 - 4.03T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 + 8.90T + 67T^{2} \) |
| 71 | \( 1 - 7.58T + 71T^{2} \) |
| 73 | \( 1 + 3.57T + 73T^{2} \) |
| 79 | \( 1 - 7.23T + 79T^{2} \) |
| 83 | \( 1 + 9.34T + 83T^{2} \) |
| 89 | \( 1 - 1.49T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.51143753953402506087216762913, −6.85640493309051195735901083012, −6.00172842185493171709357010546, −5.40817440712343855861090378198, −4.53495351515349481377675852328, −3.79416971297507135537111502832, −3.25088758180605186575832425813, −2.43441742607547720806586868812, −1.55157248212039471958871807652, 0,
1.55157248212039471958871807652, 2.43441742607547720806586868812, 3.25088758180605186575832425813, 3.79416971297507135537111502832, 4.53495351515349481377675852328, 5.40817440712343855861090378198, 6.00172842185493171709357010546, 6.85640493309051195735901083012, 7.51143753953402506087216762913
