L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 2·11-s − 13-s + 15-s − 2·17-s − 2·19-s − 2·21-s + 8·23-s + 25-s + 27-s − 6·29-s + 2·31-s − 2·33-s − 2·35-s + 2·37-s − 39-s − 2·41-s + 45-s − 6·47-s − 3·49-s − 2·51-s − 10·53-s − 2·55-s − 2·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.258·15-s − 0.485·17-s − 0.458·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.348·33-s − 0.338·35-s + 0.328·37-s − 0.160·39-s − 0.312·41-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.280·51-s − 1.37·53-s − 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.76468222511441823673679914689, −6.85660124761148147956351669883, −6.51627391485138196518082605078, −5.47861314451644308930817156280, −4.86685551301441315476996833666, −3.92314180526196010539829927341, −3.03381918139318638085341556314, −2.49994126855026978748290027566, −1.45266840813587244951522824921, 0,
1.45266840813587244951522824921, 2.49994126855026978748290027566, 3.03381918139318638085341556314, 3.92314180526196010539829927341, 4.86685551301441315476996833666, 5.47861314451644308930817156280, 6.51627391485138196518082605078, 6.85660124761148147956351669883, 7.76468222511441823673679914689