L(s) = 1 | + 2·3-s + 7-s + 9-s − 5·11-s − 8·17-s + 2·19-s + 2·21-s − 7·23-s − 4·27-s − 3·29-s − 4·31-s − 10·33-s + 37-s − 2·41-s + 3·43-s + 6·47-s + 49-s − 16·51-s − 10·53-s + 4·57-s + 4·59-s − 6·61-s + 63-s + 13·67-s − 14·69-s − 5·71-s − 6·73-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 1.94·17-s + 0.458·19-s + 0.436·21-s − 1.45·23-s − 0.769·27-s − 0.557·29-s − 0.718·31-s − 1.74·33-s + 0.164·37-s − 0.312·41-s + 0.457·43-s + 0.875·47-s + 1/7·49-s − 2.24·51-s − 1.37·53-s + 0.529·57-s + 0.520·59-s − 0.768·61-s + 0.125·63-s + 1.58·67-s − 1.68·69-s − 0.593·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.309931563221878250785917949746, −7.88106066501216715115718761865, −7.19113311152317550180120682204, −6.10814756087257490776910830132, −5.24722760241837173170631983918, −4.36540934313429224894228704464, −3.49501788588607896083881919733, −2.45050356646364018355111917792, −2.01413229267536762787594971008, 0,
2.01413229267536762787594971008, 2.45050356646364018355111917792, 3.49501788588607896083881919733, 4.36540934313429224894228704464, 5.24722760241837173170631983918, 6.10814756087257490776910830132, 7.19113311152317550180120682204, 7.88106066501216715115718761865, 8.309931563221878250785917949746