L(s) = 1 | − 3·3-s − 4·5-s − 3·7-s + 6·9-s − 3·11-s − 7·13-s + 12·15-s − 2·17-s − 2·19-s + 9·21-s − 3·23-s + 11·25-s − 9·27-s − 8·29-s − 10·31-s + 9·33-s + 12·35-s − 8·37-s + 21·39-s − 6·41-s − 11·43-s − 24·45-s + 3·47-s + 2·49-s + 6·51-s − 4·53-s + 12·55-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 1.78·5-s − 1.13·7-s + 2·9-s − 0.904·11-s − 1.94·13-s + 3.09·15-s − 0.485·17-s − 0.458·19-s + 1.96·21-s − 0.625·23-s + 11/5·25-s − 1.73·27-s − 1.48·29-s − 1.79·31-s + 1.56·33-s + 2.02·35-s − 1.31·37-s + 3.36·39-s − 0.937·41-s − 1.67·43-s − 3.57·45-s + 0.437·47-s + 2/7·49-s + 0.840·51-s − 0.549·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.93078954495451, −15.47876655732154, −14.99744735234033, −14.58344699887813, −13.36143316946967, −12.89242262598691, −12.52667901241053, −12.11753207750752, −11.65260375324399, −11.17704365273458, −10.62211124281870, −10.12313855959496, −9.668721858219296, −8.824415938563190, −8.072599920909986, −7.368031051759217, −7.051276581883740, −6.744134464035421, −5.750393784079657, −5.227364066790557, −4.811717316743150, −4.038023685978528, −3.583077188764949, −2.705408730885480, −1.703298968019941, 0, 0, 0,
1.703298968019941, 2.705408730885480, 3.583077188764949, 4.038023685978528, 4.811717316743150, 5.227364066790557, 5.750393784079657, 6.744134464035421, 7.051276581883740, 7.368031051759217, 8.072599920909986, 8.824415938563190, 9.668721858219296, 10.12313855959496, 10.62211124281870, 11.17704365273458, 11.65260375324399, 12.11753207750752, 12.52667901241053, 12.89242262598691, 13.36143316946967, 14.58344699887813, 14.99744735234033, 15.47876655732154, 15.93078954495451