L(s) = 1 | + 3·3-s − 3·5-s + 6·9-s − 11-s + 3·13-s − 9·15-s − 2·17-s − 4·19-s − 6·23-s + 4·25-s + 9·27-s − 29-s − 9·31-s − 3·33-s − 8·37-s + 9·39-s + 8·41-s − 5·43-s − 18·45-s + 7·47-s − 6·51-s − 5·53-s + 3·55-s − 12·57-s + 10·59-s − 10·61-s − 9·65-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 1.34·5-s + 2·9-s − 0.301·11-s + 0.832·13-s − 2.32·15-s − 0.485·17-s − 0.917·19-s − 1.25·23-s + 4/5·25-s + 1.73·27-s − 0.185·29-s − 1.61·31-s − 0.522·33-s − 1.31·37-s + 1.44·39-s + 1.24·41-s − 0.762·43-s − 2.68·45-s + 1.02·47-s − 0.840·51-s − 0.686·53-s + 0.404·55-s − 1.58·57-s + 1.30·59-s − 1.28·61-s − 1.11·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.948389353751108483147159522632, −7.35009738642670421514475085701, −6.68029883457391225575593494319, −5.58260551461377299797323801035, −4.33391416929939004314237841936, −3.93428194104114834475096814077, −3.40493878407245294958473105919, −2.44650034195975434628929136619, −1.64030427521635021493397399354, 0,
1.64030427521635021493397399354, 2.44650034195975434628929136619, 3.40493878407245294958473105919, 3.93428194104114834475096814077, 4.33391416929939004314237841936, 5.58260551461377299797323801035, 6.68029883457391225575593494319, 7.35009738642670421514475085701, 7.948389353751108483147159522632