L(s) = 1 | − 3-s − 4·7-s − 2·9-s + 3·11-s + 2·13-s + 17-s − 19-s + 4·21-s − 23-s + 5·27-s + 8·31-s − 3·33-s + 2·37-s − 2·39-s + 41-s − 12·43-s − 6·47-s + 9·49-s − 51-s + 4·53-s + 57-s + 12·59-s + 8·63-s − 13·67-s + 69-s + 12·71-s + 17·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s − 2/3·9-s + 0.904·11-s + 0.554·13-s + 0.242·17-s − 0.229·19-s + 0.872·21-s − 0.208·23-s + 0.962·27-s + 1.43·31-s − 0.522·33-s + 0.328·37-s − 0.320·39-s + 0.156·41-s − 1.82·43-s − 0.875·47-s + 9/7·49-s − 0.140·51-s + 0.549·53-s + 0.132·57-s + 1.56·59-s + 1.00·63-s − 1.58·67-s + 0.120·69-s + 1.42·71-s + 1.98·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.115403587253833115446082650454, −6.78875552809143481307443861423, −6.55952334758889989411856821643, −5.93260282575379282727021986575, −5.14582721655083371091910266127, −4.06987270855353481957553042192, −3.37710208741703816933090855813, −2.59407509810354169632560206350, −1.14049937280272671260589641406, 0,
1.14049937280272671260589641406, 2.59407509810354169632560206350, 3.37710208741703816933090855813, 4.06987270855353481957553042192, 5.14582721655083371091910266127, 5.93260282575379282727021986575, 6.55952334758889989411856821643, 6.78875552809143481307443861423, 8.115403587253833115446082650454