L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 0.732·7-s − 8-s + 9-s − 5.19·11-s + 12-s − 1.26·13-s − 0.732·14-s + 16-s − 0.732·17-s − 18-s + 19-s + 0.732·21-s + 5.19·22-s − 1.53·23-s − 24-s + 1.26·26-s + 27-s + 0.732·28-s + 1.19·29-s + 7.92·31-s − 32-s − 5.19·33-s + 0.732·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.276·7-s − 0.353·8-s + 0.333·9-s − 1.56·11-s + 0.288·12-s − 0.351·13-s − 0.195·14-s + 0.250·16-s − 0.177·17-s − 0.235·18-s + 0.229·19-s + 0.159·21-s + 1.10·22-s − 0.320·23-s − 0.204·24-s + 0.248·26-s + 0.192·27-s + 0.138·28-s + 0.222·29-s + 1.42·31-s − 0.176·32-s − 0.904·33-s + 0.125·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 - 7.92T + 31T^{2} \) |
| 37 | \( 1 + 4.92T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 8.19T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 5.19T + 67T^{2} \) |
| 71 | \( 1 - 0.196T + 71T^{2} \) |
| 73 | \( 1 + 7.53T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 - 0.660T + 83T^{2} \) |
| 89 | \( 1 + 9.92T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−8.192172523176057870198739897306, −7.982956563127859689007609208854, −7.13595189233618878586300889518, −6.30323596442009799741462372823, −5.24626868832154175838195164346, −4.55930675844389283910609832028, −3.22244691563603818976206701499, −2.56942227413473388111562399081, −1.56375711190024545225674481001, 0,
1.56375711190024545225674481001, 2.56942227413473388111562399081, 3.22244691563603818976206701499, 4.55930675844389283910609832028, 5.24626868832154175838195164346, 6.30323596442009799741462372823, 7.13595189233618878586300889518, 7.982956563127859689007609208854, 8.192172523176057870198739897306